LOANED   TO 
UNIVERSITY   OF    CALIFORNIA 

DEPARTMENT    OF     MECHANICAL     AND     ELECTRICAL     ENGINEERING 
FROM     PRIVATE    LIBRARY    OF 

C.    L.    CORY 

1930 


The  D.  Van  Nostrand  Company 

intend  this  book  to  be  sold  to  the  Public 
at  the  advertised  price,  and  supply  it  to 
the  Trade  on  terms  which  will  not  allow 
of  discount. 


ALTERNATING-CURRENT  MACHINES 

BEING  THE    SECOND    VOLUME   OF 

DYNAMO  ELECTRIC  MACHINERY \/, 

>       ITS   CONSTRUCTION,  DESIGN,  /,    ;      ;J/VVJ^ 
AND    OPERATION 

BY 

SAMUEL    SHELDON,  A.M.,  PH.D.,  D.Sc. 

PROFESSOR   OF   PHYSICS    AND    ELECTRICAL    ENGINEERING    AT    THE    POLYTECHNIC 
INSTITUTE   OF   BROO1 

INSTITUTE   OF    ELECTR 

HOBART 

ASSOCIATE   OF    THE    AMERICA/f  INSTITUTE  OF.AJECTRICAL    ENGINEERS 


N,  B.S.  E.E. 


EERING     AT     THE     POLYTECHNIC 
ASSOCIATE    OF    THE    AMERICAN 
ELECTRICAL   ENGINEERS 


EDITION 

COMPETEL  Y  REWRITTEN 


NEW   YORK 

D.  VAN    NOSTRAND    COMPANY 
23  MURRAY  AND  27  WARREN  STS. 

LONDON 

CROSBY   LOCKWOOD   &   SON 

7  STATIONERS'  HALL  COURT,  LUDGATE  HILL 

1908 


Engineering 
Library 


COPYRIGHT,  1908,  BY 
D.  VAN   NOSTRAND   COMPANY 


Stanhope  Jpres* 

F.   H     3ILSON    COMPANTI 
BOSTON.    U.S.A. 


PREFACE   TO    FIRST    EDITION. 


THIS  book,  like  its  companion  volume  on  Direct  Current 
Machines,  is  primarily  intended  as  a  text-book  for  use  in 
technical  educational  institutions.  It  is  hoped  and  be- 
lieved that  it  will  also  be  of  use  to  those  electrical,  civil, 
mechanical,  and  hydraulic  engineers  who  are  not  perfectly 
familiar  with  the  subject  of  Alternating  Currents,  but  whose 
work  leads  them  into  this  field.  It  is  furthermore  intended 
for  use  by  those  who  are  earnestly  studying  the  subject 
by  themselves,  and  who  have  previously  acquired  some 
proficiency  in  mathematics. 

There  are  several  methods  of  treatment  of  alternating- 
current  problems.  Any  point  is  susceptible  of  demonstra- 
tion by  each  of  the  methods.  The  use  of  all  methods  in 
connection  with  every  point  leads  to  complexity,  and  is 
undesirable  in  a  book  of  this  character.  In  each  case  that 
method  has  been  chosen  which  was  deemed  clearest  and 
most  concise.  No  use  has  been  made  of  the  method  of 
complex  imaginary  numbers. 

A  thorough  understanding  of  what  takes  place  in  an 
alternating-current  circuit  is  not  to  be  easily  acquired.  It 
is  believed,  however,  that  one  who  has  mastered  the  first 
four  chapters  of  this  book  will  be  able  to  solve  any  practi- 
cal problem  concerning  the  relations  which  exist  between 
power,  electro-motive  forces,  currents,  and  their  phases  in 

789556 


iv  PREFACE. 

series  or  multiple  alternating-current  circuits  containing 
resistance,  capacity,  and  inductance. 

The  next  four  chapters  are  devoted  to  the  construction, 
principle  of  operation,  and  behavior  of  the  various  types  of 
alternating-current  machines.  Only  American  machines 
have  been  considered. 

A  large  amount  of  alternating-current  apparatus  is  used 
in  connection  with  plants  for  the  long-distance  transmission 
of  power.  This  subject  is  treated  in  the  ninth  chapter. 
The  last  chapter  gives  directions  for  making  a  variety  of 
tests  on  alternating-current  circuits  and  apparatus. 

No  apology  is  necessary  for  the  introduction  of  cuts  and 
material  supplied  by  the  various  manufacturing  companies. 
The  information  and  ability  of  their  engineers,  and  the  taste 
and  skill  of  their  artists,  are  unsurpassed,  and  the  informa- 
tion supplied  by  them  is  not  available  from  other  sources. 
For  their  courteous  favors  thanks  is  hereby  given. 


PREFACE   TO   THE    SEVENTH   EDITION. 


THE  extensive  adoption  of  this  volume  as  a  text-book 
for  the  use  of  students  on  other  than  electrical  courses  and 
the  growing  tendency,  in  many  Institutions,  to  require 
more  thorough  and  extended  work  in  electrical  subjects 
from  such  students,  have  determined  the  scope  of  the 
present  revision.  In  those  cases  where  insufficient  time  is 
available  for  covering  all  the  ground  contained  herein,  it 
will  be  found  that  portions,  which  the  instructor  will 
probably  desire  to  omit,  are  so  treated  that  the  remainder 
will  constitute  a  coordinated  treatment.  It  is  also  believed 
that,  in  the  majority  of  Institutions,  the  book  as  a  whole 
will  be  found  adapted  for  the  use  of  students  on  electrical 
courses.  The  manner  of  presentation  is  in  many  parts 
different  from  that  which  would  be  employed  in  a  book 
written  for  engineers,  but  an  extended  experience  in 
teaching  young  men  of  average  attainments  has  proved  it 
to  be  effective.  As  a  student  seldom  gets  a  thorough 
understanding  of  a  subject  of  this  character  without 
making  numerical  computations,  problems  have  been 
introduced  at  the  conclusion  of  each  chapter. 


CONTENTS. 


CHAPTER    I 
PROPERTIES  OF  ALTERNATING  CURRENTS. 

ART  PAGE 

1.  Definition  of  an  Alternating  Current i 

2.  Frequency i 

3.  Wave-Shape 3 

4.  Distortion 5 

5.  Effective  Values  of  E.M.F.  and  of  Current 7 

6.  Form  Factor  of  Non-Sine  Curves 10 

7.  Phase 12 

8.  Power  in  Alternating-Current  Circuits 14 

9.  Non-Sine  Waves 18 

10.    E.M.F.'s  in  Series 20 

Problems 25 


CHAPTER    II. 

SELF-INDUCTION. 

11.  Self-Inductance 26 

12.  Unit  of  Self-Inductance 27 

13.  Practical  Values  of  Inductances 29 

14.  Things  which  influence  the  Magnitude  of  L 30 

15.  Formulae  for  calculating  Inductances 31 

16.  Growth  of  Current  in  an  Inductive  Circuit 33 

17.  Decay  of  Current  in  an  Inductive  Circuit 34 

18.  Magnetic  Energy  of  a  started  Current 36 

19.  Current  produced  by  a  Harmonic  E  M.F.  in  a  Circuit  having  Resist- 

ance and  Inductance 37 

20.  Instantaneous  Current  produced  by  a  Harmonic  E.M.F.  in  a  Circuit 

having  Resistance  and  Inductance 41 

21.  Choke  Coils 44 

Problems 47 

vii 


viii  CONTENTS. 


CHAPTER    III. 

CAPACITY- 
ART.  PACK 

22.  Condensers 48 

23.  Capacity  Formulae 53 

24.  Connection  of  Condensers  in  Parallel  and  in  Series 55 

25.  Decay  of  Current  in  a  Condensive  Circuit 57 

26.  Energy  stored  in  Dielectric 60 

27.  Condensers  in  Alternating-Current  Circuits  —  Hydraulic  Analogy.  60 

28.  Phase  Relations 61 

29.  Current  and  Voltage  Relations 63 

30.  Instantaneous  Current  in  a  Circuit  having  Capacity  and  Resistance  65 
Problems .68 


CHAPTER    IV. 
ALTERNATING-CURRENT  CIRCUITS. 

31.  Resistance,  Inductance,  and  Capacity  in  an  Alternating-Current 

Circuit ' .  70 

32.  Definitions  of  Terms 71 

33.  Representation     of     Impedance    and     Admittance    by    Complex 

Numbers 74 

34.  Instantaneous  Current  in  a  Circuit  having  Inductance,  Capacity  and 

Resistance 76 

35.  Resonance 80 

36.  Damped  Oscillations 82 

37.  Polygon  of  Impedances 83 

38.  A  Numerical   Example  applying  to  the  Arrangement  shown   in 

Fig.  50 85 

39.  Polygon  of  Admittances 87 

40.  Impedances  in  Series  and  in  Parallel 90 

Problems 92 

CHAPTER    V. 

ALTERNATORS. 

41.  Alternators 94 

42.  Electromotive  Force  generated 96 

43.  Armature  Windings 99 

44.  Voltage  and  Current  Relations  in  Two-phase  Systems 101 

45.  Voltage  and  Current  Relations  in  Three-phase  Systems 104 


CONTENTS.  IX 

ART  PAGE 

46.  Voltage  and  Current  Relations  in  Four-Phase  Systems 106 

47.  Measurement  of  Power 107 

48.  Saturation 113 

49.  Regulation 116 

50.  E.M.F.  and  M.M.F.  Methods  of  calculating  Regulation 119 

51.  Regulation  for  Constant  Potential 124 

52.  Efficiency 133 

53.  Rating 135 

54.  inductor  Alternators 136 

55.  Revolving  Field  Alternators 139 

56.  Self- Exciting  Alternators 145 

Problems .  . . 146 

CHAPTER    VI. 
THE  TRANSFORMER. 

57.  Definitions 149 

58.  The  Ideal  Transformer 151 

59.  Core  Flux 154 

60.  Transformer  Losses 1 56 

61.  Core  Losses 156 

62.  Exciting  Current 159 

63.  Equivalent  Resistance  and  Reactance  of  a  Transformer 163 

64.  Copper  Losses 165 

65.  Efficiency 166 

66.  Calculation  of  Equivalent  Leakage  Inductance 168 

67.  Regulation 173 

68.  Circle  Diagram 179 

69.  Methods  of   connecting  Transformers 181 

70.  Lighting  Transformers 188 

71.  Cooling  of  Transformers 192 

72.  Constant  Current  Transformers 195 

73.  Polyphase  Transformers 198 

Problems 200 

CHAPTER    VII. 

MOTORS. 
INDUCTION  MOTORS. 

74.  Rotating  Field 202 

75.  The  Induction  Motor 203 

76.  Starting  of  Squirrel-Cage  Motors 207 


X  CONTENTS. 

ART,  PAGE 

77.  Principle  of  Operation  of  the  Induction  Motor 210 

78.  Relation  between  Speed  and  Efficiency 213 

79.  Determination  of  Torque 214 

80.  The  Transformer  Method  of  Treatment 215 

81.  Leakage  Reactance  of  Induction  Motors 216 

82.  Calculation  of  Exciting  Current 227 

83.  Circle  Diagram  by  Calculation 231 

84.  Circle  Diagram  by  Test 233 

85.  Performance  Curves  from  Circle  Diagram 236 

86.  Method  of  Test  with  Load 238 

87.  Phase  Splitters 241 

88.  The  Single-Phase  Induction  Motor 242 

89.  The  Monocyclic  System  .  „ 244 

90.  Frequency  Changers 244 

91.  Speed  Regulation  of  Induction  Motors 245 

92.  The  Induction  Wattmeter 246 

SYNCHRONOUS  MOTORS. 

93.  Synchronous  Motors 249 

94.  Special  Case 252 

95.  The  Motor  E.M.F 256 

96.  Starting  Synchronous  Motors 258 

97.  Parallel  Running  of  Alternators 262 

SINGLE-PHASE  COMMUTATOR  MOTORS. 

98.  Single- Phase  Commutator  Motors 262 

99.  Plain  Series  Motor 264 

100.  Characteristics  of  Plain  Series  Motor 268 

101.  Compensated  Series  Motors 270 

102.;  Sparking  in  Series  Motors .  .  .  „ 274 

103.  Repulsion  Motors „ 277 

104.  Series-Repulsion  Motor 280 

Problems .  .c 282 

CHAPTER    VIII. 

CONVERTERS. 

105.  The  Converter , 284 

106.  E.M.F.  Relations 286 

107.  Current  Relations 288 

108.  Heating  of  the  Armature  Coils 290 

109.  Capacity  of  a  Converter . «, 291 


CONTENTS.  XI 

ART.  PAGE 

i  io.    Starting  a  Converter 291 

in.    Armature  Reaction 291 

112.  Regulation  of  Converters 293 

113.  Mercury  Vapor  Converter 296 

Problems    299 


CHAPTER    IX. 
POWER  TRANSMISSION. 

114.  Superiority  of  Alternating  Currents 300 

115.  Frequency 302 

116.  Number  of  Phases 304 

117.  Voltage 305 

1 18.  Economic  Drop 306 

119.  Line  Resistance 309 

1 20.  Line  Inductance .  .  . 310 

121.  Line  Capacity 313 

122.  Regulation 316 

123.  Conductor  Material 318 

124.  Insulators 319 

125.  Sag  of  Conductors 322 

126.  Line  Structure 326 

127.  Spans  and  Layout 329 

128.  Example  of  Design  of  Transmission  Line 331 

Problem 343 


ALTERNATING-CURRENT  MACHINES. 


CHAPTER    I. 

PROPERTIES  OF  ALTERNATING  CURRENTS. 

1.  Definition   of   an   Alternating  Current.  —  An    alter- 
nating current  of  electricity  is  a  current  which  changes 
its    direction    of    flow    at    regularly    recurring    intervals. 
Between    these    intervals  the  value  of   the  current    may 
vary  in  any  way.      In  usual  practice,  the  value  varies  with 
some  regularity  from  zero  to  a  maximum,  and  decreases 
with  the  same  regularity  to  zero,  then  to  an  equal  max- 
imum in  the  other  direction,  and  finally  to  zero  again.     In 
practice,  too,  the  intervals  of  current  flow  are  very  short, 
ranging  from  ^  to  ^|7  second. 

2.  Frequency.  —  When,  as  stated  above,  a  current  has 
passed  from  zero  to  a  maximum  in  one  direction,  to  zero, 
to  a  maximum  in  the  other  direction,  and  finally  to  zero 
again,  it  is  said  to  have  completed  one  cycle.     That  is  to 
say,  it  has  returned  to  the  condition  in  which  it  was  first 
considered,   both  as  to  value  and  as  to  direction,  and  is 
prepared  to  repeat  the  process  described,  making  a  second 
cycle.     It  should  be  noted  that  it  takes  two  alternations 
to  make  one  cycle.     The  tilde  ( ~ )  is  frequently  used  to 
denote  cycles. 


2  ALTERNATING-CURRENT    MACHINES. 

The  term  frequency  is  applied  to  the  number  of  cycles 
completed  in  a  unit  time,  i.e.,  in  one  second.  Occasionally 
the  word  alternations  is  used,  in  which  case,  unless  other- 
wise specified,  the  number  of  alternations  per  minute  is 
meant.  Thus  the  same  current  is  spoken  of  as  having  a 
frequency  of  25,  or  as  having  3000  alternations.  The  use 
of  the  word  alternations  is  condemned  by  good  practice. 
In  algebraic  notation  the  letter  f  usually  stands  for  the 
frequency. 

The  frequency  of  a  commercial  alternating  current 
depends  upon  the  work  expected  of  it.  For  power  a 
low  frequency  is  desirable,  particularly  for  converters. 
The  great  Niagara  power  plant  uses  a  frequency  of  25. 
Lamps,  however,  are  operated  satisfactorily  only  on  fre- 
quencies of  50  or  more.  Early  machines  had  higher 
frequencies, —  125  and  133  (16,000  alternations)  being 
usual,  — but  these  are  almost  entirely  abandoned  because 
of  their  increased  losses  and  their  unadaptability  to  the 
operation  of  motors  and  similar  apparatus. 

In  the  Report  of  the  Committee  on  Standardization  of 
the  American  Institute  of  Electrical  Engineers  is  the 
following:  "In  alternating-current  circuits,  the  follow- 
ing frequencies  are  standard: 

25  ~ 
60 

"These  frequencies  are  already  in  extensive  use,  and 
it  is  deemed  advisable  to  adhere  to  them  as  closely  as 
possible." 

The  frequency  of  an  alternating  current  is  always  that 
of  the  E.M.F.  producing  it.  To  find  the  frequency  of  the 
pressure  or  the  current  produced  by  any  alternating-cur- 


PROPERTIES   OF   ALTERNATING   CURRENTS.         3 

rent   generator,  if    V  be  the  number  of   revolutions   per 
minute,  and  /  be  the  number  of  pairs  of  poles,  then 


3.   Wave-shape If,    in    an    alternating    current,    the 

instantaneous  values  of  current  be  taken  as  ordinates,  and 
time  be  the  abscissae,  a 
curve,  as  in  Fig.  I,  may  be 
developed.  The  length  of 
the  abscissa  for  one  com- 
plete cycle  is—  seconds. 

Imagine  a  small  cylinder, 
Fig.  2,  carried  on  one  end  of  a  wire,  and  rotated  uniformly 
about  the  other  end  in  a  vertical  plane.  Imagine  a  hori- 
zontal beam  of  parallel  rays  of  light  to  be  parallel  to  the 
plane  of  rotation,  and  to  cast  a  shadow  of  the  cylinder  on 


Fig.   i. 


Fig.  2. 

a  plane  screen  perpendicular  to  the  rays.  The  shadow 
will  move  up  and  down,  passing  from  the  top  of  its  travel 
to  the  bottom  in  a  half  revolution,  and  from  the  bottom 


4  ALTERNATING-CURRENT   MACHINES. 

back  to  the  top  in  another  half  revolution  with  a  perfect 
harmonic  motion.  Now  imagine  the  screen  to  be  moved 
horizontally  in  its  own  plane  with  a  uniform  motion,  and 
the  positions  of  the  shadow  suitably  recorded  on  it,  — as 

on  sensitized  paper  or  on 
a  photographic  film,  a 
slotted  screen  protecting 
all  but  the  desired  portion 
from  exposure.  Then  the 
trace  of  the  shadow  will 
be  as  in  Fig.  3.  The 
abscissas  of  this  curve 

may  be  taken  as  time,  as  in  the  preceding  curve,  the  ab- 
scissa of  one  complete  cycle  being  the  time  in  seconds  of 
one  revolution.  Or,  with  equal  relevancy,  the  abscissae 
may  be  expressed  in  degrees.  Consider  the  cylinder  to  be 
in  a  zero  position  when  the  radius  to  which  it  is  attached 
is  horizontal.  Then  the  abscissa  of  any  point  is  the  angle 
which  must  be  turned  through  in  order  that  the  cylinder 
may  cast  its  shadow  at  that  point.  In  this  case  the  abscissa 
of  a  complete  cycle  will  be  360°,  or  2  TT.  Consideration  of 
the  manner  in  which  the  curve  has  been  formed  shows 
that  the  ordinate  of  any  point  is  proportional  to  the  sine 
of  the  abscissa  of  that  point,  expressed  in  degrees.  Hence 
this  is  called  a  sinusoid  or  sine  curve. 

If  the  maximum  ordinate  of  this  curve,  which  corresponds 
to  the  length  of  the  moving  radius,  or  OA  in  Fig.  4,  repre- 
sents Em,  then  the  instantaneous  value  of  the  voltage,  E', 
at  /  seconds  after  the  beginning  of  any  cycle,  will  be  AB,  or 
Em  sin  0.  But,  since  OA  traverses  2  n  radians  during  one 
complete  revolution,  it  will  sweep  over  2  TT/  radians  per 
second,  and,  as  angular  velocity,  represented  by  a>,  is  the 


PROPERTIES    OF   ALTERNATING   CURRENTS.        5 

angle  turned  through  in  unit  time,  it  follows  that  the  angular 
velocity  of  OA  is  2  xf.  The  angular  velocity  may  also  be 
expressed  as  0/t,  or  0  =  cot 
=  2  njt. 

Hence         £'  =  Em  sin  cot 

=  Em  sin  2  TT//, 

which  is  equivalent  to  neglecting 

all  those  intervals  of  time  cor-  Fj 

responding  to  whole  cycles,  and 

considering  only  the  time  elapsed  since  the  end  of  the  last 

completed    cycle.     In    Fig.    4,   OA    is  termed   the   radius 

vector,  and  0,  the  vectorial  angle  or  displacement.     Graphic 

solutions  of  alternating-current  problems  may  be  effected 

by  the  use  of  vectors. 

4.  Distortion.  —  The  ideal  pressure  curve  from  an  alter- 
nator is  sinusoidal.  Commercial  alternators,  however,  do 
not  generate  true  sinusoidal  pressures.  But  the  sine  curve 
can  be  treated  with  relative  simplicity,  and  the  curves  of 
practice  approximate  so  closely  to  the  sine  form,  that  mathe- 
matical deductions  based  on  sine  curves  can  with  propriety 
be  applied  to  those  of  practice.  Two  of  these  actual  curves 
are  shown  in  Fig.  5. 

The  shape  of  the  pressure  curve  is  affected  by  irregular 
distribution  of  the  magnetic  flux.  Also  uneven  angular 
velocity  of  the  generator  will  distort  the  wave-shape, 
making  it,  relative  to  the  true  curve,  lower  in  the  slow 
spots  and  higher  in  the  fast  ones.  Again,  the  magnetic 
reluctance  of  the  armature  may  vary  in  different  angular 
positions,  particularly  if  the  inductors  are  laid  in  a  few 
large  slots.  This  would  cause  a  periodic  variation  in  the 


6 


ALTERNATING-CURRENT    MACHINES. 


reluctance  of  the  whole  magnetic  circuit  and  a  correspond- 
ing pulsation  of  the  total  magnetic  flux.  All  these  influ- 
ences operate  at  open  circuit  as  well  as  under  load. 


E.M.F.  CURVE 
3  PHASE 
40  POLE 
2000  K.W. 


E.M.F.  CURVE 
SINGLE  PHASE 

8  POLE 
500  WATTS 

125  ^ 
NOT  LOADED 


Fig.  5. 

There  are  two  other  causes  which  act  to  distort  the 
wave-shape  only  when  under  load.  For  any  separately 
excited  generator,  a  change  in  the  resistance  or  apparent 
resistance  of  the  external  circuit  will  cause  a  change  in  the 


PROPERTIES  OF  ALTERNATING  CURRENTS.   / 

terminal  voltage  of  the  machine.  As  is  explained  later, 
the  apparent  resistance  (impedance)  of  a  circuit  to  alter- 
nating currents  depends  upon  the  permeability  of  the  iron 
adjacent  to  the  circuit.  Permeability  changes  with  mag- 
netization. Now,  because  an  alternating  current  is  flow- 
ing, the  magnetization  changes  with  the  changing  values 
of  current.  This,  by  varying  the  permeability,  sets  up  a 
pulsation  in  the  impedance  and  affects  the  terminal  volt- 
age of  the  machine,  periodically  distorting  the  wave  of 
pressure  from  the  true  sine. 

There  are  cases  of  synchronously  pulsating  resistances. 
The  most  common  is  that  of  the  alternating  arc.  With 
the  same  arc  the  apparent  resistance  of  the  arc  varies  in- 
versely as  the  current.  So  when  operated  by  alternating 
currents,  the  resistance  of  a  circuit  of  arc  lamps  varies  syn- 
chronously, and  distorts  the  pressure  wave-shape  in  a 
manner  analogous  to  the  above/ 

Summing  up,  the  wave-shape  of  pressure  may  be  dis- 
torted :  At  open  circuit  as  well  as  under  load ;  by  lack  of 
uniformity  of  magnetic  distribution,  by  pulsating  of  mag- 
netic field,  by  variation  in  angular  velocity  of  armature  ; 
and  under  load  only  ;  by  pulsation  of  impedance,  by  pulsa- 
tion of  resistance.  And  the  effects  of  any  or  all  may  be 
superimposed. 

5.  Effective   Values   of  E.M.F.  and  of  Current One 

ampere  of  alternating  current  is  a  current  of  such  instan- 
taneous values  as  to  have  the  same  heating  effect  in  a  con- 
ductor as  one  ampere  of  direct  current.  This  somewhat 
arbitrary  definition  probably  arose  from  the  fact  that  alter- 
nating currents  were  first  commercially  employed  in  light- 
ing circuits,  where  their  utility  was  measured  by  the  heat 


8  ALTERNATING-CURRENT   MACHINES. 

they  produced  in  the  filaments ;  and  further  from  the  fact 
that  the  only  means  then  at  hand  of  measuring  alternating 
currents  were  the  hot-wire  instruments  and  the  electro- 
dynamometer,  either  of  which  gives  the  same  indication 
for  an  ampere  of  direct  current  or  for  what  is  now  called 
an  ampere  of  alternating  current. 

The  heat  produced  in  a  conductor  carrying  a  current  is 
proportional  to  the  square  of  the  current.  In  an  alternat- 
ing current,  whose  instantaneous  current  values  vary,  the 
instantaneous  rate  of  heating  is  not  proportional  to  the 
instantaneous  value,  nor  yet  to  the  square  of  the  average 
of  the  current  values,  but  to  the 
square  of  the  instantaneous  cur- 
rent value.  And  so  the  average 
heating  effect  is  proportional  to 
~~  the  mean  of  the  squares  of  the 
\  instantaneous  currents. 

Fi£-  6-  The  average  current  of  a  sinu- 

soidal wave  of  alternating  current,  whose  maximum  value 
is  7m,  is  equal  to  the  area  of  one  lobe  of  the  curve,  Fig.  6, 
divided  by  its  base  line  TT.  Thus 


Im  sin  BdB 

/*j  u  -*  m  r 

i»  =  =  —  f—  COS  i 


r 

Jo 

But  the  heating  value  of  such  a  current  varies,  as 

C" 

72  =  A-! =  ^ [°-  -  I  sin  2  0T  =  -7m2. 

7T  7T     \_2  4  Jo  2 

The  square  root  of  this  quantity  is  called  the  effective 
value  of  the  current,  /  =  — ^-     This  has  the  same  heating 

•V/o 


— 
V2 


PROPERTIES    OF   ALTERNATING   CURRENTS.        9 

effect  as  a  direct  current  /,  and  the  effective  values  are 
always  referred  to  unless  expressly  stated  otherwise. 
Alternating-current  ammeters  are  designed  to  read  in 
effective  amperes. 

Since  current  is  dependent  upon  the  pressure,  the 
resistance  or  apparent  resistance  of  a  circuit  remain- 
ing constant,  it  is  obvious  that  if  /  =  — ^  then  does 

E  2       Va 

also  E  =  — ™-    Likewise  if  average  /  =  -  Im  then  does  also 

V2  w 

average  E  =  -  Em.     Or  these  may  be  demonstrated  in  a 

7T 

manner  analogous  to  the  above. 

The  maximum  value  of  pressure  is  frequently  referred 
to  in  designing  alternator  armatures,  and  in  calculating 
dielectric  strength  of  insulation.  There  have  arisen  vari- 
ous ways  of  indicating  that  effective  values  are  meant, 
for  instance,  the  expressions,  sq.  root  of  mean  sq.,  V/, 
Vmean  square.  In  England  the  initials  R.M.S.  are  fre- 
quently used  for  root  mean  square. 

„,,         ,.     Effective  E.M.F.    .         „    ,     ,        . 
The  ratio  — —        —  is    called   the  form-factor, 

Average  E.M.F. 

since    its    value    depends    upon 
the  shape  of  the  pressure  wave. 
For  the  curve  Fig.  7,  the  form- 
factor  is  unity.     As  a  curve  be- 
comes  more    peaked,,  its    form-  Fi«-  ?• 
factor  increases,  due  to  the  superior  weight  of  the  squares 
of  the  longer  ordinates. 

In  the  sinusoid  the  values  found  above  give 
i 

— —  77 

Form-factor  = =  i.n. 

'-£. 


10 


ALTERNATING-CURRENT    MACHINES. 


6.  Form  Factor  of  Non-Sine  Curves.  —  For  the  deter- 
mination of  the  form  factor,  three  methods  may  be  used, 
according  to  the  character  of  the  wave  shape.  First,  if  the 
equation  of  the  curve  is  known,  the  analytical  method  may 
be  employed.  For  example,  take  the  ellipse,  Fig.  8.  Its 

b    / ~ 

equation  is  y  =  -  v  2  ax  —  or. 


The  average  ordinate  is 


/»2« 
/ 
Jo 


_ 
y  dx       -   I      '\/2  ax  —  x2  dx 


bVx  -  a     /—        — 2        fl2         _j  *!2a       6 /a2     \ 

-  V  2  ajc  —  x    +  —  vers  - 1  —  TT  ) 

a[_     2 2 a  Jo        a  \2_ / 


ab 

2 


and  since  a  =  -  this  becomes  —  •  The  square  of  the  mean 
ordinate  is 


r*  2, 

/      yzdx 
Ji 


^;    ?[  ^    *r 

x2)dx      —  lax2  —  — 
a?\_  3  Jo 


7T  3    7T 

but  a  =  -    hence  this  becomes  —  and  it  follows  that  the 

2  ,-  3 

effective  value  is  V  —  b. 
V3 


PROPERTIES   OF   ALTERNATING   CURRENTS.      II 

b        4\/2_ 


Therefore  the  form  factor  = 


1=  1.04. 


4_a 


Second,  the  geometrical  method  may  be  used  in  calculating 
the  form  factor  of  simple  wave  shapes,  as  for  example,  Fig.  9. 

area  of  Fig.  9     b  [a  +  2  a]     3  , 

The  average  ordmate  =  -  -  -  &_^_i  -  -  = 

base 

,_.-,..  t  /volume  of  Fig.  10 

The  effective  value     =  v  -  ;  --  r~      — 

base  line 

The  volume  of  Fig.  10  is 

2  ab2  +  2  .  J  ab2  =  b2  (2  a  +  f  a)  =  f  a 


/ 

/ 
/               \ 

b                      \ 

—  «—  *U  2 

.-_  H        ^J 

Fig.  9. 

Hence  the  effective  value  is 
~ab2 


40, 


,     /-       T    i      r         ,.  . 

=  ^  V  f  and  the  form  factor  is 


4\/2 


And  third,  the  form  factor  of  irregular  curves,  as  for 
example  the  lower  E.M.F.  curve  of  Fig.  5,  may  be  deter- 
mined by  the  use  of  a  planimeter.  The  average  value 

= &i_5  _  ^0  g^     *po  obtain  the  effective  value. 

base 

a  curve  of  squared  ordinates  must  be  plotted.  The  area  of 
this  curve  divided  by  its  base  is  the  mean  ordinate  and  the 
square  root  of  this  mean  square  is  the  effective  value  of  the 
voltage,  which  for  the  curve  in  question  is  .685  Em.  Hence 

.68s 

->-  1.14. 


the  form  factor  = 


.60 


12 


ALTERNATING-CURRENT   MACHINES. 


Probably  no  alternators  give  sine  waves,  but  they  ap- 
proach it  so  nearly  that  the  value  i.n  can  be  used  .in  most 
calculations  without  sensible  error. 

7.    Phase The  curves  of  the  pressure  and  the  current 

in  a  circuit  can  be  plotted  together,  with  their  respective 
ordinates  and  common  abscissae,  as  in  Fig.  n.  In  some 

cases  the  zero  and  the 
maximum  values  of  the 
current  curve  will  occur 
at  the  same  abscissae  as 
do  those  values  of  the 
pressure  curve,  as  in  Fig. 
ii.  In  such  a  case  the 

current  is  said  to  be  in  phase  with  the  pressure.  In  other 
cases  the  current  will  reach  a  maximum  or  a  zero  value  at 
a  time  later  than  the  corresponding  values  of  the  pressure, 
and  since  the  abscissae  are  indifferently  time  or  degrees, 
the  condition  is  represented  in  Fig.  12.  In  such  a  case, 
the  current  is  said  to  be  ottt  of  phase  with,  and  to  lag  be- 
hind the  pressure.  In 

Still     Other     cases     the  /          x*-\~\     LAGGING  CURRENT 

curves  are  placed  as  in 
Fig.  13,  and  the  current 
and  pressure  are  again 
out  of  phase,  but  the 

current  is  said  to  lead  Fig-  I2- 

the  pressure.  The  distance  between  the  zero  ordinate  of 
one  sine  curve  and  the  corresponding  zero  ordinate  of 
another,  may  be  measured  in  degrees,  and  is  called  the 
angular  displacement  or  phase  difference.  This  angle  of 
lag  or  of  lead  is  usually  represented  by  <£.  When  one 


PROPERTIES    OF  ALTERNATING   CURRENTS.      13 


curve  has  its  zero  ordinate  coincident  with  the  maximum 
ordinate  of  the  other,  as  in  Fig.  14,  there  is  a  displacement 
of  a  quarter  cycle  (<£  =  90°),  and  the  curves  are  said  to  be 
at  right  angles.  This 

term  owes  its  origin  to  X/^X^    \\LEADINGCURRENT 

the  fact  that  the  radii 
whose  projections  will    ~£ 
trace  these  curves,  as 
in    §  3,    are   at    right 
angles  to  each    other.  Flg-  I3< 

If  the  zero  ordinates  of  the  two  curves  coincide,  but  the 
positive  maximum  of  one  coincides  with  the  negative  maxi- 
mum of  the  other,  as  in 
Fig.  15,  then  <£  =  180°, 
and  the  curves  are  in  op- 
posite phase. 

An  alternator  arranged 
to  give  a  single  pressure 
wave  to  a  two-wire  circuit  is 


RIGHT  ANGLES 


Fig.   14. 

and    the    current 


OPPOSITE  PHASE 


said  to  be  a  single  phaser, 
in  the  circuit  a  single-phase  current. 
Some  machines  are  arranged  to  give  pressure  to  two  dis- 
tinct circuits  —  each  of 
which,  considered  alone, 
is  a  single-phase  circuit 
—  but  the  time  of  maxir 

0  \ 

mum  pressure  in  one  is         \ 
the  time  of   zero  pres- 
sure   in    the    other,    so 
that  simultaneous  pres- 
sure  'curves    from    the    two 


Fig.  15. 

circuits    take    the    form    of 


Fig.  1 6.      Such  is  said  to  be  a  two-phase  or  quarter-phase 


ALTERNATING-CURRENT    MACHINES. 


system,  and  the  generator  is  a  two-phaser.  A  three-phase 
system  theoretically  has  three  circuits  of  two  wires  each. 
The  maximum  positive  pressure  on  any  circuit  is  displaced 
f roni  that  of  either  of  the  other  circuits  by  1 20°.  As  the 

algebraic  sum  of  the  cur- 
rents in  all  these  circuits 
(if  balanced)  is  at  every  in- 
stant equal  to  zero,  the 
three  return  wires,  one  on 


TWO  PHASE 

Fig.  16. 


each  circuit,  may  be  dis- 
pensed with,  leaving  but 
three  wires.  The  three  sim- 
ultaneous curves  of  E.M.F. 
are  shown  in  Fig.  17.  The  term  polyphase  applies  to  any 
system  of  two  or  more  phases.  An  «-phase  system  has  n 
circuits  and  n  pressures  with  successive  phase  differences 

of  - —  degrees. 
n 

8.  Power  in  Alternating-Current  Circuits —  With  a  direct- 
current  circuit,  the  power  in  the  circuit  is  equal  to  the 
product  of  the  pressure  in  volts  by  the  current  strength  in 
amperes.  In  an  alternating- 
current  circuit,  the  instan- 
taneous power  is  the  product 
of  the  instantaneous  values 
of  current  strength  and 
pressure.  If  the  current 
and  pressure  are  out  of 
phase  there  will  be  some 
instants  when  the  pressure  will  have  a  positive  value  and 
the  current  a  negative  value  or  vice  versa.  At  such  times 
the  instantaneous  power  will  be  a  negative  quantity,  i.e., 


PROPERTIES    OF   ALTERNATING   CURRENTS.      15 

power  is  being  returned  to  the  generator  by  the  disappear- 
ing magnetic  field  which  had  been  previously  produced  by 
the  current.  This  condition  is  shown  in  Fig.  18,  where 
the  power  curve  has  for  its  ordinates  the  product  of  the 
corresponding  ordinates  of  pressure  and  current.  These 
are  reduced  by  multiplying  by  a  constant  so  as  to  make 
them  of  convenient  size. 
The  circuit,  therefore, 
receives  power  from  the 
generator  and  gives  power 
back  again  in  alternating 
pulsations  having  twice 
the  frequency  of  the  gen- 
erator. It  is  clear  that 
the  relative  magnitudes  Fig*  l8' 

of  the  negative  and  positive  lobes  of  the  power  curve  will 
vary  for  different  values  of  <£,  even  though  the  original 
curves  maintain  the  same  size  and  shape.  So  it  follows 
that  the  power  in  an  alternating-current  circuit  is  not 
merely  a  function  of  E  and  /,  as  in  direct-current  circuits, 
but  is  a  function  of  E,  /,  and  <£,  and  the  relation  is  deduced 
as  follows :  — 

Let  the  accent  (')  denote  instantaneous  values.     If  the 
current  lag  by  the  angle  <f>,  then  from  §  3, 

Er  =  Em  sin  a, 
where,  for  convenience, 

a  =  2  IT  ft, 

and  -Tr  =  fm  sin  (a  —  <£). 

Remembering  that 

TP  T 

E  =  — p,    and    /  =  — p=  (§5)  the  instantaneous  power, 

V2  V2 

P'  =  E' I'  =  2EI  sin  a  sin  (a  —  <£). 


16  ALTERNATING-CURRENT   MACHINES. 

But  sin  (a  —  <£)  =  sin  a  cos  <f>  —  cos  a  sin  <£, 

so  /"=  2  ^"/(sin2  a  cos  <£  —  sin  a  cos  a  sin  <£. 


Remembering  that   <£  is  a  constant,   the  average  power 
over  1  80°, 


j>r«.  2^/sin  <£  f"  . 

I     sin2  cu/a  --  -  I    Sin  a  cos  cu/a 

Jo  TT  Jo 

<^ri          i    .        >      2£Ss'md>\~i    .    ,   I"" 
-a  --  Sin  2  a      --  -  sin2  a     . 

L2  4  Jo  7T  L2  JO 


P  =  EIcos  < 


Should  the  current  lead  the  pressure  by  <£°,  then  the 
leading  equation  would  be 

Pr  =•  2  Efsin  a  sin  (a  +  <£), 
which  gives  the  same  expression, 

P  =  jg'/COS  <£, 

which  is  the  general  expression  for  power  in  an  alternating- 
current  circuit. 

The  above  may  also  be  shown  by  the  use  of  vectors.  Let 
OA  and  OB  of  Fig.  19  rep- 
resent the  effective  values  of 
E.M.F.  and  current  respect- 
ively, taking  the  former  as  the 
datum  line  and  assuming  the 
latter  to  lag  <j>  degrees  behind 
Fig.  19.  the  E.M.F.  The  line,  OB, 

may  be  resolved  into  two  com- 
ponents, one  along  OA  and  the  other  at  right  angles  to  it. 
These  components,  OP  and  OQ,  are  termed  respectively 
the  power  and  wattless  components  of  the  current.  The 
actual  power  expended  in  the  circuit  is  OA  X  OP  =  El  cos  (f> 


PROPERTIES   OF  ALTERNATING   CURRENTS.      17 

and  the  wattless  power,  or  that  alternately  supplied  to  and 
received  from  the  circuit,  is  OA  X  OQ  =  El  sin  (j>. 

Since,  to  get  the  true  power  in  the  circuit,  the  apparent 
power,  or  volt-amperes,  must  be  multiplied  by  cos  <£,  this 
quantity  is  called  the  power  factor  of  the  circuit.  If  the 
pressure  and  current  are  in  phase,  <j)  =  o°,  and  the  power 
factor  is  unity. 

It  is  important,  at  this  point,  to  consider  the  graphical 
method  of  addition  or  subtraction  of  vector  quantities,  a 
process  which  is  frequently  employed  in  the  treatment  of 
alternating-current  circuits.  Let  A  and  B,  Fig.  20,  be  two 
lines  whose  lengths  and  whose  directions  respectively  repre- 
sent the  magnitudes  and  time  or  space  locations  of  two  vector 


Fig.  20. 

quantities.  These  maybe  E.M.F.'s,  or  currents  as  the  case 
may  be.  The  sum  of  two  vectors  is  given  in  magnitude  and 
in  direction  by  the  concurrent  diagonal  of  a  parallelogram 
the  adjacent  sides  of  which  represent  the  vectors  in  size  and 
direction.  To  substract  one  quantity  from  another  vec- 
torially,  it  is  but  necessary  to  change  its  sign  and  add  it  to 
the  other.  Representing  vectorial  addition  by  the  symbol 
0 ,  and  vectorial  subtraction  by  0 ,  the  results  of  the  various 
additions  and  subtractions  of  A  and  B  become  evident  from 
the  figure. 


18  ALTERNATING-CURRENT   MACHINES. 

9.  Non-Sine  Waves.  —  As  sine  waves  of  E.M.F.  or  cur- 
rent are  seldom  obtained  in  practice,  it  is  convenient,  in 
accurate  calculations,  to  refer  to  their  equivalent  sine  waves. 
An  equivalent  sine  wave  is  one  having  the  same  frequency 
and  the  same  mean  effective  value  as  the  given  wave.  Con- 
sider two  non-sine  waves,  one  of  E.M.F.  and  the  other  of 
current,  their  zero  or  maximum  values  being  displaced  by  an 
angle  (f>n.  The  phase  difference  of  these  two  non-sine  waves 
cannot  be  considered  as  the  angle  0B,  but  is  that  phase  dis- 
placement of  their  equivalent  sine  waves  which  would  give 
the  same  average  of  the  instantaneous  power  values  as  the 


Fig.  2 


non-sine  waves.  Therefore,  to  find  the  phase  displacement  of 
two  non-sine  waves,  it  is  necessary,  first,  to  plot  a  power  curve 
and  determine  its  average  ordinate,  Par;  second,  to  calculate 
the  mean  effective  values  of  the  curves,  represented  respec- 
tively by  E  and  /;  and  third,  to  substitute  these  values  in 
the  equation  Pav  =  El  cos  <£  from  which  (j>  can  be  obtained. 
In  general,  it  can  be  said,  that  when  the  form  factors  of 
both  waves  are  less  than  that  of  the  sine  curve,  their  phase 


PROPERTIES   OF  ALTERNATING   CURRENTS.      19 

difference  is  greater  than  the  displacement  of  their  zero  or 
maximum  values;  and  likewise,  if  their  form  factors  exceed 
the  value  i.n,  then  the  phase  displacement  is  less  than  the 
displacement  of  corresponding  values  of  the  curves. 

As  a  numerical  example:  Find  the  phase  displacement 
of  two  semi-circular  waves,  having  their  zero  values  one- 
twelfth  of  a  cycle  or  30°  apart.  Let  one  be  a  pressure  curve, 
whose  maximum  value  is  120  volts,  and  the  other,  a  current 
curve,  whose  maximum  value  is  6  amperes.  These  are 
shown  in  Fig.  21. 

The  average  ordinate  of  the  power  curve,  determined  by 
subtracting  the  negative  area  from  the  positive  and  then 
dividing  by  the  base,  is  found  to  be  383  watts.  The 


effective  value  of  each  curve  is  */  £2 1  y*  for  the  circle 

being  2  bx  —  x2  where  b  is  the  maximum  ordinate. 

C"  C* 

2b  I     oc doc  —  I    x*dx 

't/O  *J  Q 


Since  b  =  - ,  this  becomes  — ,    and  hence    the    effective 
2  6 

value  is  —= -  •     From  the  relations 
V6 

7T          7T  f        r  i       TC          7t 

-  :  —  =6:1    and    -  :  — _  =  120  :  E, 
2    V6  2    V6 

it  follows  that  the  effective  values  of  current  and  voltage 
are  respectively  4.9  amperes  and  98  volts.  Then  383  =  98  X 
4.9  cos  <f>9  which  gives  as  the  phase  displacement,  $  =  37°, 
instead  of  30°. 


20 


ALTERNATING-CURRENT   MACHINES. 


io.    E.M.F.'s    in    Series.  —  Alternating    E.M.F.'s    that 
may    be  put  in  series   may  differ  in    magnitude,   in   fre- 
quency,   in    phase    relation, 
and    in    form    or    shape    of 
wave. 

If  two  harmonic  E.M.F.'s 
of  the  same  frequency  and 
phase  be  in  series,  the  re- 
sulting E.M.F.  is  merely 
the  sum  of  the  separate 
E.M.F.'s.  This  condition  is 
shown  in  Fig.  22,  in  which 
the  two  E.M.F.'s  are  plotted 
together,  and  the  resulting  E.M.F.  plotted  by  making  its 
instantaneous  values  equal  to  the  sum  of  the  correspond- 
ing instantaneous  values  of  the  component  E.M.F.'s.  The 
maximum  of  the  resultant  E.M.F.  is  evidently 


Fig.  22. 


and  since 


and 


V2 


as  was  stated. 

If  two  E.M.F.'s  of  the  same  frequency,  but  exactly 
opposite  in  phase,  be  placed  in  series,  it  may  be  similarly 
shown  that  the  resultant  E.M.F.  is  the  numerical  differ- 
ence of  the  component  E.M.F.'s.  This  case  may  occur  in 
the  operation  of  motors. 

The  most  general  case  that  occurs  is  that  of  a  number 
of  alternating  E.M.F.'s  of  the  same  frequency,  but  of 


PROPERTIES   OF   ALTERNATING   CURRENTS.      21 

different  magnitudes  and  phase  displacements.  The 
changes  in  magnitude  and  phase  and  the  phase  relation  of 
the  resulting  curve  of  E.M.F.  are  shown  in  Fig.  23,  where 
recourse  is  had  once  again  to  the  harmonic  shadowgraph. 
But  two  components,  El  and  E^  are  treated,  whose  phase 
displacement  is  <£L.  The  radii  vectors  Eim  and  E^m  are 
laid  off  from  o  with  the  proper  angle  <#>1  between  them, 
and  the  shadows  traced  by  their  extremities  are  shown  in 
the  dotted  curves.  The  instantaneous  value  of  the  result- 
ant E.M.F.  is  the  algebraic  sum  of  the  corresponding  in- 


Fig.  23. 

stantaneous  values  of  the  component  E.M.F.'s,  and  the 
resultant  curve  of  E.M.F.  is  traced  in  the  figure  by  the 
solid  line.  But  this  solid  curve  is  also  the  trace  of  the  ex- 
tremity of  the  line  Ena  which  is  the  vector  sum  (the  result- 
ant of  the  force  polygon)  of  the  component  pressures,  Eim 
and  E2m.  This  is  evident  from  the  fact  that  any  instan- 
taneous value  of  the  resultant  pressure  curve  is  the  sum  of 
the  corresponding  instantaneous  values  of  the  component 
curves,  or  (§  3) 

Er  =  Elm  sin  <o/  -+-  EZn  si 


Again  from  the  force  polygon 

E^  sin  (wt  -\-  <£)  =  £lm  sin  o>/  -f-  E2m  sin  (<o/  -f- 


22 


ALTERNATING-CURRENT   MACHINES. 


Hence  at  any  instant 

E'  =  Em  sin  (o>/  +  <£), 

wherefore  the  extremity  of  the  line  Em  traces  the  curve  of 
resultant  pressure,  <£  being  its  angular  displacement  from 
Ev.  If  a  third  component  E.M.F.  is  to  be  added  in  series, 
it  may  be  combined  with  the  resultant  of  the  first  two  in 
an  exactly  similar  manner. 

So  it  may  be  stated  as  a  general  proposition,  that  if  any 
number  of  harmonic  E.M.F'  's,  of  the  same  frequency,  but 
of  various  magnitudes  and 
phase  displacements,  be 
connected  in  series,  the 
resulting  harmonic  E.M.F. 
will  be  given  in  magnitude 
and  phase  by  the  vector  sum  of  the  component  E.M.F's. 
The  analytic  expressions  for  E  and  <£  may  be  derived  by 
inspection  of  the  diagram,  and  are 


Fig.  24- 


E  = 


and 


tan  < 


+ 


Fig.  25. 


As  a  numerical  example,  suppose  three  alternators,  Fig. 
24,  to  be  connected  in  series.  Suppose  these  to  give  sine 
waves  of  pressure  of  values  2^=  70,  ^a  =  6o,  and  ^  =  40 


PROPERTIES    OF   ALTERNATING   CURRENTS       23 


volts  respectively.  Considering  the  phase  of  El  to  be 
the  datum  phase,  let  the  phase  displacements  be  <£x  =  o°, 
(j)^  =  40°,  and  (j>3  =  75°,  respectively.  It  is  required  to  find 
E  and  <j).  Completing  the  parallelograms  or  completing 
the  force  polygon  as  shown  in  Fig.  25,  it  is  found  that 
E  =  148.7  volts  and  <£  =  32.1°. 

Alternating  E.M.F.'s  of  different  frequencies  in  series 
will  give,  in  general,  an  irregular  wave  form.  In  practice, 
the  frequencies  of  some  E.M.F.'s  are  multiples  of  the  fre- 
quency of  another,  called  the  fundamental  E.M.F.,  or  first 
harmonic.  The  pressure  curve  having  twice  this  frequency 
is  termed  the  second  harmonic;  another  having  three  times 
this  frequency,  the  third  harmonic,  and  so  on.  The  result- 
ant instantaneous  E.M.F.  is  obtained  by  adding  the  pressure 


Fig.  26. 


values  of  all  the  components  at  that  instant.    It  is  expressed 


'  =  El 


sin  (2  cat 


4-  E3m  sin 
+  Enm  sin 


9^2)  +  ••• 


24  ALTERNATING-CURRENT    MACHINES. 

where  <pv  ^>2,  .  .  .  ,  <£n_i,  are  the  phase  differences  between 
Elm  and  E2m,  Elm  and  E3m,  .  .  .  ,  Elm  and  Enm  respectively 
when  sin  cot  =  o. 

When  both  odd  and  even  harmonics  are  present,  the 
resulting  curve  will  have  unlike  lobes,  but  when  only  odd 
harmonics  occur,  as  is  usual  in  electrical  machinery,  the 
lobe  above  the  horizontal  axis  and  the  other  below  it  will 
be  similar.  Fig.  26  shows  the  resulting  E.M.F.  of  three 
harmonic  components  for  the  values,  Elm  =  100  volts, 
E2m  =  40  volts,  E3m  =  20  volts,  <£j  =  30°  and  <£2  =  45°. 

Let  it  be  required  to  find  Ef,  2j  seconds  after  the  beginning 
of  a  cycle,  the  frequency  of  £t  being  25  -**. 

2  7T 

CUt  =  2  7T  25.  2j  =    Il6f  7T  Or  -    • 
O 


3  w/  =  3  .  —  =  2  TT  or  o,  then  (3  cut  +  <£2)  =  —  • 

Hence       Er  =  100  sin  —  ^  +  40  sin  -  --  \-  20  sin  — 
3  2  4 

=  86.6  —  40  +  14.1  =  60.7 

volts,  which  agrees  with  the  value  of  the  ordinate  at  120°, 
in  the  figure. 

When  the  resulting  pressure  curve  is  given,  it  is  possible, 
by  graphical  and  analytical  methods,  to  determine  which 
harmonics  are  present,  their  maximum  values,  and  phase 
displacements. 


PROBLEMS.  25 


PROBLEMS. 

1.  What  must  be  the  speed  of  a   i2-pole  alternator  to  yield  an 
E.M.F.  of  60  cycles  ? 

2.  Find  the  instantaneous  current  value  in  a  circuit,  in  which  a 
25 ~**  alternating  current  of  70.7  amperes  flows,  6.0066  seconds  after  the 
completion  of  a  cycle. 

3.  How  many  amperes  flow  in  a  circuit,  when  the  instantaneous 
value  of  the  current  is  5  amperes,  30°  after  the  beginning  of  a  cycle  ? 

4.  What  is  the  frequency  of  an  E.M.F.  which  assumes  its  effective 
value  every  .01  second  ? 

5.  Find  the  average  and  effective  values  of  a  semi-circular  wave 

shape,  the  maximum  value  being  —  *       Determine  form  factor. 

6.  Find  the  form  factor  of  a  triangular  wave  shape. 

7.  Determine  the  form  factor  of  the  upper  curve  of  Fig.  5. 

8.  Find  the  instantaneous  voltage  produced  by  a  50^  alternator," 
generating  parabolic  waves  of  120  volts  maximum  value,  if  seconds 
after  the  beginning  of  a  cycle. 

9.  What  is  the  phase  displacement  between  E  and  /,  respectively 
of  100  volts  and  10  amperes  maximum  value,  when  the  power  in  the 
circuit  is  424  watts  ? 

10.  Find  the  phase  difference  of  two  non-sine  waves  of  voltage  and-1 
current,  whose  zero  values  are  45°  apart.     Let  the  wave  shape  be  as 
shown  in  Fig.  9  and  let  Em  =  100  volts  and  Im  =  10  amperes. 

11.  Four  60^  alternators,  generating  respectively  100,  80,  90  and 
50  volts,  are  connected  to  a  circuit.     What  will  be  the  value  of  the  result- 
ing pressure,  and  what  will  be  its  phase  with  respect  to  that  of  the  100 
volts,  if  the  phase  difference  between  successive  components  is  45°? 

12.  If  the  three  E.M.F.'s,  shown  in  Fig.  26,  are  impressed  upon  a 
circuit,  what  will  be  the  resulting  instantaneous  voltage  5.015  seconds 
after  the  beginning  of  a  cycle? 


26  ALTERNATING-CURRENT   MACHINES. 


CHAPTER    II. 

SELF-INDUCTION. 

ii.  Self-Inductance. — The  subject  of  inductance  was 
briefly  treated  of  in  §  15,  vol.  i.,  of  this  work ;  but,  since  it 
is  an  essential  part  of  alternating-current  phenomena,  it 
will  be  discussed  more  fully  in  this  chapter.  When  lines 
of  force  are  cut  by  a  conductor  an  E.M.F.  is  generated  in 
that  conductor  (§  13,  vol.  i.).  A  conductor  carrying  cur- 
rent is  encircled  by  lines  of  force.  When  the  current  is 
first  started  in  such  a  conductor,  these  lines  of  force  must 
be  established.  In  establishing  itself,  each  line  is  con- 
sidered as  having  cut  the  conductor,  or,  what  is  equivalent, 
been  cut  by  the  conductor.  This  notion  of  lines  of  force 
is  a  convenient  fiction,  designed  to  render  an  understand- 
ing of  the  subject  more  easy.  To  account  for  the  E.M.F. 
of  self-induction,  the  encircling  lines  must  be  considered 
as  cutting  the  conductor  which  carries  the  current  that 
establishes  them,  during  their  establishment.  It  may  be 
considered  that  they  start  from  the  axis  of  the  conductor 
at  the  moment  of  starting  the  current  in  the  circuit ;  that 
they  grow  in  diameter  while  the  current  is  increasing ;  that 
they  shrink  in  diameter  when  the  current  is  decreasing; 
and  that  all  their  diameters  reduce  to  zero  upon  stopping 
the  current.  At  any  given  current  strength  the  conductor 
is  surrounded  by  many  circular  lines,  the  circles  having 
various  diameters.  Upon  decreasing  the  strength  those  of 


SELF-INDUCTION.  2/ 

smaller  diameter  cut  the  conductor  and  disappear  into  a 
point  on  the  axis  of  the  conductor  previous  to  the  cutting 
by  those  of  larger  diameter.  The  number  of  lines  accom- 
panying a  large  current  is  greater  than  the  number  accom- 
panying a  smaller  current. 

The  E.M.F.  of  self-induction  is  always  a  counter  E.M.F. 
By  this  is  meant  that  its  direction  is  such  as  to  tend  to 
prevent  the  change  of  current  which  causes  it.  When  the 
current  is  started  the  self-induced  pressure  tends  to  oppose 
the  flow  of  the  current  and  prevents  its  reaching  its  full 
value  immediately.  When  the  circuit  is  interrupted  the 
E.M.F.  of  self-induction  tends  to  keep  the  current  flowing 
in  the  same  direction  that  it  had  originally. 

12.  Unit  of  Self-Inductanc3. — The  selj-inductance,  or 
the  coefficient  of  s elj '-induction  of  a  circuit  generally  rep- 
resented by  L  or  /,  is  that  constant  by  which  the  time 
rate  of  change  of  the  current  in  a  circuit  must  be  multi- 
plied in  order  to  give  the  E.M.F.  induced  in  that  circuit. 
Its  absolute  value  is  numerically  equal  to  the  number  of 
lines  of  force  linked  with  the  circuit,  per  absolute  unit  of 
current  in  the  circuit,  as  is  shown  below.  By  linkages,  or 
number  of  lines  linked  with  a  circuit,  is  meant  the  sum 
of  the  number  of  lines  surrounding  each  portion  of  the 
circuit.  For  instance,  a  coil  of  wire  consisting  of  ten 
turns,  and  threaded  completely  through  by  twelve  lines 
of  force,  is  said  to  have  1 20  linkages. 

The  absolute  unit  of  self -inductance  is  too  small  for 
ordinary  purposes,  and  a  practical  unit,  the  henry,  is  used. 
This  is  io9  times  as  large  as  the  c.  G.  s.  or  absolute  unit. 

The  Paris  electrical  congress  of  1900  adopted  as  the 
unit  of  magnetic  flux  the  maxwell,  and  of  flux  density  the 


28  ALTERNATING-CURRENT    MACHINES. 

gauss.  A  maxwell  is  one  line  of  force.  A  gauss  is  one 
line  of  force  per  square  centimeter.  If  a  core  of  an  electro- 
magnet has  a  transverse  cross-section  of  30  sq.  cm.,  and  is 
uniformly  permeated  with  60,000  lines  of  force,  such  a 
core  may  be  said  to  have  a  flux  of  60,000  maxwells  and  a 
flux  density  of  2000  gausses. 

In  §  13,  vol.  i.,  it  has  been  shown  that  the  pressure  gene- 
rated in  a  coil  of  wire  when  it  is  cut  by  lines  of  force  is 


where  n  is  the  number  of  turns  in  a  coil,  and  where  e  is 
measured  in  c.  G.  s.  units,  &  in  maxwells,  and  /  in  seconds. 
In  a  simple  case  of  self-induction  the  maxwells  set  up  are 
due  solely  to  the  current  in  the  conductor.  Now  let  K  be 
a  constant,  dependent  upon  the  permeability  of  the  mag- 
netic circuit,  such  that  it  represents  the  number  of  max- 
wells set  up  per  unit  current  in  the  electric  circuit  ;  then, 
indicating  instantaneous  values  by  prime  accents, 

&  =  Kir, 

and  d&  =  Kdi. 

The  E.M.F.  of  self-induction  may  then  be  written 

di 

e>=-Knm 

By   the   definition   of    the  coefficient    of    self-induction, 
whose  c.  G.  s.  value  is  represented  by  /, 


From  the  last  two  equations,  it  is  seen  that  /  =  Kn.  Kn  is 
evidently  the  number  of  linkages  per  absolute  unit  current. 
The  negative  sign  indicates  that  the  pressure  is  counter 
E.M.F. 


SELF-INDUCTION.  29 

In  practical  units, 

P  Tdl  • 

Es=~L~dt 

A  circuit  having  an  inductance  of  one  henry  will  have  a 
pressure  of  one  volt  induced  in  it  by  a  uniform  change  of 
current  of  one  ampere  per  second. 

13.  Practical  Values  of  Inductances. — To  give  the 
student  an  idea  of  the  values  of  self-inductance  met  with  in 
practice,  a  number  of  examples  are  here  cited. 

A  pair  of  copper  line  wires,  say  a  telephone  pole  line, 
will  have  from  two  to  four  milhenrys  (.002  to  .004  henrys) 
per  mile,  according  to  the  distance  between  them,  the 
larger  value  being  for  the  greater  distance. 

The  secondary  of  an  induction  coil  giving  a  2"  spark  has 
a  resistance  of  about  6000  ohms  and  50  henrys. 

The  secondary  of  a  much  larger  coil  has  30,000  ohms 
and  about  2000  henrys. 

A  telephone  call  bell  with  about  75  ohms  has  1.5  henrys. 

A  coil  found  very  useful  in  illustrative  and  quantitive 
experiments  in  the  alternating-current  laboratory  is  of  the 
following  dimensions.  It  is  wound  on  a  pasteboard  cylinder 
with  wooden  ends,  making  a  spool  8.5  inches  long  and  2 
inches  internal  diameter.  This  is  wound  to  a  depth  of  1.5 
inch  with  No.  16  B.  and  S.  double  cotton-covered  copper 
wire,  there  being  about  3000  turns  in  all.  A  bundle  of 
iron  wires,  16  inches  long,  fits  loosely  in  the  hole  of  the 
spool.  The  resistance  of  the  coil  is  10  ohms,  and  its  in- 
ductance without  the  core  is  0.2  henry.  With  the  iron 
core  in  place  and  a  current  of  about  0.2  ampere,  the  induc- 
tance is  about  1.75  henrys.  This  coil  is  referred  to  again 
in  §  16. 


30  ALTERNATING-CURRENT   MACHINES. 

The  inductance  of  a  spool  on  the  field  frame  of  a  gene- 

rator is  numerically 

< 


where  <£  is  the  total  flux  from  one  pole,  n  the  number 
of  turns  per  spool,  and  If  the  field  current  of  the  machine. 
It  is  evident  that  the  value  of  L  may  vary  through  a  wide 
range  with  different  machines. 

14.  Things  Which  Influence  the  Magnitude  of  L.  —  If  all 

the  conditions  remain  constant,  save  those  under  considera- 
tion, then  the  self-inductance  of  a  coil  will  vary  :  directly  as 
the  square  of  the  number  of  turns  ;  directly  as  the  linear 
dimension  if  the  coil  changes  its  size  without  changing  its 
shape  ;  and  inversely  as  the  reluctance  of  the  magnetic 
circuit. 

Any  of  the  above  relations  is  apparent  from  the  follow* 
ing  equations.  The  numerical  value  of  the  self  -induc- 
tance is 

7  ^ 

/=  n  —  • 
i 

As  shown  in  Chapter  2,  vol.  i., 

M.M.F.    _  4  irni 


_ 

~ 


reluctance  ~~      c 


where  c  is  the  mean  length  in  centimeters  of  the  magnetic 
circuit,  A  its  mean  cross-sectional  area  in  square  centi- 
meters, and  /A  is  permeability. 

Then,  if  (R  stand  for  the  reluctance, 

n    4  icni  „    A       4ir«2 

/  =  7~  ^^T^^T' 

PA 
which  is  independent  of  t. 


SELF-INDUCTION.  31 

If,  as  is  generally  the  case,  there  is  iron  in  the  magnetic 
circuit,  it  is  practically  impossible  to  keep  p  constant  if  any 
of  the  conditions  are  altered  ;  and  it  is  to  be  particularly 
noted,  that  with  iron  in  the  magnetic  circuit,  L  is  by  no 
means  independent  of  /. 

15.  Formulas  for  Calculating  Inductances.  —  Circle: 
For  a  cylindrical  conductor  of  radius  r  cm.  and  length  /  cm., 
bent  into  a  circle  and  surrounded  with  a  medium  of  unit 
permeability  the  self-inductance  in  henries  is 


L  =  10 


This  is  accurate  to  within  0.2  %  when  the  radius  of  the  circle 
is  greater  than  ten  times  that  of  the  cylindrical  conductor. 

Straight  Wire:  For  a  straight  cylindrical  conductor  of 
radius  r  cm.  and  length  /  cm.  in  a  medium  of  unit  permea- 
bility the  self-inductance  in  henries  is 


L  =  io-9 


Parallel  Wires:  For  a  return  circuit  of  two  parallel  cylin- 
drical wires  of  radius  r  cm.,  d  cm.  apart  from  center  to  center, 
each  of  permeability  ft  and  of  /  cm.  length,  the  self-inductance 
in  henries  is 


L=  io-9 


•KM 


Solenoids:   The  formula  given  below  for  the  self-induct- 
ance of  a  solenoid  of  any  number  of  layers  will  give  results 


32 


ALTERNATING-CURRENT   MACHINES. 


accurate  to  within  one-half  of  a  per  cent  even  for  short 
solenoids,  where  the  length  is  only  twice  the  diameter,  the 
accuracy  increasing  as  the  length  increases. 


IV  4  < 
(m  —  2)  a'' 


X 


Where 

w  is  the  number  of  layers, 

aQ  is  the  mean  radius  of  the  solenoid, 

au  av  as>  -  •  -  am  are    the   mean  radii    of    the  various 

layers, 

/  is  the  length  of  the  solenoid, 
da    is    the    radial    distance    between    two    consecutive 

layers, 
n  is  the  number  of  turns  per  unit  length. 

A  simple  and  convenient  formula  for  the  calculation  of 
the  self  -inductance  of  a  single-layer  solenoid  is  as  follows  : 


2  2  a 
L  =  A7i:2n2\  —  = 


8 


/4  a2  +  P      3 


where   a  is  the  mean  radius  and  /  is  the  length  of  the 
solenoid. 


SELF-INDUCTION.  33 

The  natural  logarithms  used  in  preceding  formulae  can  be 
obtained  by  multiplying  the  common  logarithm  of  the  num- 
ber, the  mantissa  and  characteristic  being  included,  by  2.3026. 

The  inductance  of  all  circuits  is  somewhat  less  for 
extremely  high  frequencies  than  for  low  ones. 

1  6.  Growth  of  Current  in  an  Inductive  Circuit.  —  If  a 
constant  E.M.F.  be  applied  to  the  terminals  of  a  circuit 
having  both  resistance  and  inductance,  the  current  does 
not  instantly  assume  its  full  ultimate  value,  but  logarith- 
mically increases  to  that  value. 

At  the  instant  of  closing  the  circuit  there  is  no  current 
flowing.  Let  time  be  reckoned  from  this  instant.  At 
any  subsequent  instant,  t  seconds  later,  the  impressed 
E.M.F.  may  be  considered  as  the  sum  of  two  parts,  E^ 
and  Er.  The  first,  E^  is  that  part  which  is  opposed  to, 
and  just  neutralizes,  the  E.M.F.  of  self-induction,  so  that 


but 


The  second  part,  Er,  is  that  which  is  necessary  to  send 
current  through  the  resistance  of  the  circuit,  according  to 
Ohm's  Law,  so  that 

Er  =  RL 

If  the  impressed  E.M.F. 

dt 
then  (41  -  RI)  dt  =  Ldl, 

an  dt  =  E  -  RI  dl  =  ~  R  *  E  -  RI  " 


34 


ALTERNATING-CURRENT   MACHINES. 


Integrating  from  the  initial   conditions  t  =  o,  /=o  to  any 
conditions  t  =  t,  /=/', 

L 

Rt 


and 


E      *t 


where  e  is  the  base  of  the  natural  system  of  logarithms. 

This  equation  shows  that  the  rise  of  current  in  such  a 
circuit  is  along  a  logarithmic  curve,  as  stated,  and  that  when 
/  is  of  sufficient  magnitude  to 

_R 

render  the  term  c  L  negli- 
gible the  current  will  follow 
Ohm's  Law,  a  condition  that 
agrees  with  observed  facts. 

Fig.  27  shows  the  curve  of 
growth  of  current  in  the  coil 
referred  to  in  § 1 3.    The  curve 
is   calculated  by  the    above   formula 
noted. 

The  ratio  —  is  called  the  time  constant  of  the  circuit, 
R 

for  the  greater  this  ratio  is,  the  longer  it  takes  the  current 
to  obtain  its  full  ultimate  value. 

17.  Decay  of  Current  in  an  Inductive  Circuit.  —  If  a  cur- 
rent be  flowing  in  a  circuit  containing  inductance  and  re- 
sistance, and  the  supply  of  E.M.F.  be  discontinued, 
without,  however,  interrupting  the  continuity  of  the  circuit, 
the  current  will  not  cease  instantly,  but  the  E.M.F.  of 


the    conditions 


SELF-INDUCTION.  35 

self-induction  will  keep  it  flowing  for  a  time,  with  values 
decreasing  according  to  a  logarithmic  law. 

An  expression  for  the  value  of  this  current  at  any  time, 
/  seconds  after  cutting  off  the  source  of  impressed 
E.M.F.,  may  be  obtained  as  in  the  preceding  section.  Let 
time  be  reckoned  from  the  instant  of  interruption  of  the 
impressed  E.M.F.  The  current  at  this  instant  may  be 

represented  by  — ,  and  is  due  solely  to  the  E.M.F.  of  self- 
induction. 

Therefore  £  =  RI  ,  ^  —  =  o 

dt 

or  RI  =  —  L  — 

dt  ' 

.-.,//=  _  ^  ^  . 
R    I 

Integrating  from  the  initial  conditions  t  =  o,  /=  -— ,  to 
the  conditions,  /  =  /,/=/', 

C*  L    CIfdl 

Jo  R  J  E^    I 

R 

L          /' 
^ 


DECAYING  CURRENT 

E.M.F.-O  R 


.02   .03  .04   .05    .06    .07  .08   .09     Tl  j  Tf 

and  1     = 


-=  ~ 


SECONDS  a 

Fig.  28. 

which  is  seen  to  be  the  term  that  had  to  be  subtracted  in 
the  formula  for  growth  of  current.  This  shows  clearly 
that  while  self-induction  prevents  the  instantaneous  attain- 
ment of  the  normal  value  of  current,  there  is  eventually  no 
loss  of  energy,  since  what  is  subtracted  from  the  growing 
current  is  given  back  to  the  decaying  current. 

Fig.  28  is  the  curve  of  decay  of  current  in  the  same  cir- 


36  ALTERNATING-CURRENT   MACHINES. 

cuit  as  was  considered  in  Fig.  27.  The  ordinates  of  the 
one  figure  are  seen  to  be  complementary  to  those  of  the 
other. 

18.  Magnetic  Energy  of  a  Started  Current.  —  If  a  cur- 
rent /  is  flowing  under  the  pressure  of  E  volts,  the  power 
expenditure  is  El  watts,  and  the  work  performed  in  the 
interval  of  time  dt\§ 


During  the  time  required  to  establish  a  steady  flow  of 
current  after  closing  the  circuit,  the  impressed  E  may  be 
considered  as  made  up  of  two  parts,  one,  Er,  required  to  send 
/'  through  the  resistance  of  the  circuit,  and  the  other,  Es, 
which  opposes  the  E.M.F.  of  self-induction.  Erl  dt  appears 
as  heat,  while  ESI  dt  is  stored  in  the  magnetic  field.  Since 


.dW=-LIdI. 

Integrating  through  the  full  range,  from  o  to  W  and  from 
oto/, 


/MF  fl 

I      dW  =  -L    I    Idl. 

*J  0  **  Q 


which  is  an  expression  for  the  work  done  upon  the  magnetic 
field  in  starting  the  current.  When  the  current  is  stopped 
the  work  is  done  by  the  field,  and  the  energy  is  returned  to 
the  circuit. 

The  formula  assumes  the  value  of  L  to  be  constant  during 
the  rise  and  fall  of  the  current,  but  this  is  not  the  case  with 
an  iron  magnetic  circuit.  If  L  is  taken  as  the  average  of 


SELF-INDUCTION.  37 

the  instantaneous  values  of  self-inductance  between  the 
limiting  values  of  the  current,  then  the  formula  for  the 
energy  stored  in  the  field  still  holds  true. 

Since  iron  has  always  a  hysteretic  loss,  some  of  the 
energy  is  consumed,  and  the  work  given  back  at  the  dis- 
appearance of  the  field  is  less  than  that  used  to  establish 
the  field  by  the  amount  consumed  in  hysteresis. 

19.  Current  Produced  by  a  Harmonic  E.M.F.  in  a  Cir- 
cuit Having  Resistance  and  Inductance.  —  Given  a  circuit 
of  resistance  R  and  inductance  L  upon  which  is  impressed 
a  harmonic  E.M.F.  E  of  frequency  /,  to  find  the  current 
/  in  that  circuit. 

Represent  by  w  the  quantity  2irf. 

At  any  instant  of  time,  /,  let  the  instantaneous  value  of 
the  current  be  /'. 

To  maintain  this  current  requires  an  E.M.F.  whose  value 
at  this  instant  is  I' R.  Represent  this  by  E'r. 

From  §  3,  in  a  harmonic  current, 

/'  =  Im  sin  co/, 
hence,  Erf  =  Rim.  s*n  <*>*• 

Evidently  Er'  has  its  maximum  value  RIm  =  Erm  at  <o/  =  9o° 
or  270°,  and  its  effective  value  is  Er  =  RI. 

The  counter  E.M.F.  of  self-induction  at  the  same  instant 
of  time,  /,  is 

*'— Z^- 

A~       '   dt 

But  as  before,  /'=  /m  sin  <•>/, 

so  dlf  =  <*>Im  cos  CD/  dt, 

and  £,'  =  —  <*LIm  cos  o>/. 


ALTERNATING-CURRENT    MACHINES. 


Evidently  £,'  has  a  maximum  value  of   —  o>Z/m  =E9m  at 
tat  =  o°  or  1 80°,  and  its  effective  value 

Es  =  -  <oZ7. 

It  is  clear  that  the  impressed  E.M.F.  must  be  of  such 
a  value  as  to  neutralize  Es  and  also  supply  Er.  But  these 
two  pressures  cannot  be  simply 
added,  since  the  maximum  value  of 
one  occurs  at  the  zero  value  of  the 
other;  that  is,  they  are  at  right 
angles  to  each  other,  as  defined  in 
§  7.  Reference  to  Fig.  29  will 


Fig.  29. 


make  it  clear  that  combining  these  at  right  angles  will 
give  as  a  resultant  the  pressure  \IE?  4-  E? ;  and  it  is  this 
pressure  that  the  impressed  E.M.F.  E  must  equal  and 
oppose.  So 

£  = 


from  which 


(o>Z/)2, 


7= 


This  is  a  formula  which  must  be  used  in  place  of  Ohm's 
Law  when  treating  inductive  circuits  carrying  harmonic 
currents.  It  is  evident  that,  if  the  inductance  or  the  fre- 
quency be  negligibly  small  (direct  current  has  f  =  o),  the 
formula  reduces  to  Ohm's  Law ;  but  for  any  sensible  val- 
ues of  <o  and  L  the  current  in  the  circuit  will  be  less  than 
that  called  for  by  Ohm's  Law. 

The  expression  V/i?2  4-  a>2Z2  is  called  the  impedance  of 
the  circuit,  and  also  the  apparent  resistance.  The  term  R 
is  of  course  called  resistance,  while  the  term  <>>L,  which  is 
2  -nfLy  is  called  the  reactance.  Both  are  measured  in  ohms. 

The  effective  value  of  the  counter  E.M.F.  of  self-indue- 


SELF-INDUCTION.  39 

tion  can  be  determined  as  follows,  without  employing  the 
calculus  ;  that  it  must  be  combined  at  right  angles  with 
RI  is  not  directly  evident.  Disregarding  the  direction  of 
flow,  an  alternating  current  /  reaches  a  maximum  value  im 
2f  times  per  second.  The  maximum  number  of  lines  of 
force  linked  with  the  circuit  on  each  of  these  occasions  is 
lim.  The  interval  of  time,  from  when  the  current  is  zero 
with  no  linkages,  to  when  the  current  is  a  maximum  with 

lim  linkages,  is  —  j  second.      The  average  rate  '  of  cutting 
47 

lines,  then,  is  —  -  ,  and  is  equal  to  the  average  E.M.F.  of 

4/ 

self-induction  during  the  interval.  It  has  the  same  value 
during  succeeding  equal  intervals  ;  i.e., 


4? 

The  effective  value  is  (§  5)  therefore, 

es  =  —  2  Tcfli  =  (t>fi, 
and  in  practical  units, 

Es  =  -  2  trfLI. 

Since  the  squares  of  the  quantities  R,  L,  and  <o  enter 
into  the  expression  for  the  impedance,  if  one,  say  R,  is 
moderately  small  when  compared  with  L  or  w,  its  square 
will  be  negligibly  small  when  compared  with  Z2  or  o>2.  The 
frequency,  because  it  is  a  part  of  w,  may  be  a  considerable 
factor  in  determining  the  impedance  of  a  circuit. 

Having  recourse  once  again  to  the  harmonic  shadow- 
graph described  in  §  3,  the  phase  relation  between  im- 
pressed E.M.F.  and  current  may  be  made  plain.  It  has 
already  been  shown  that  Er  and  E8  are  at  right  angles  to 


40  ALTERNATING-CURRENT    MACHINES. 

each  other.  Since  the  pressure  Er  is  the  part  of  the  im- 
pressed E.M.F.  which  sends  the  current,  the  current  must 
be  in  phase  with  it.  Therefore  there  is  always  a  phase 
displacement  of  90°  between  /  and  Es.  This  relation  is 
also  evident  from  a  consideration  of  the  fact  that  when  / 
reaches  its  maximum  value  it  has,  for  the  instant,  no  rate 
of  change;  hence  the  flux,  which  is  in  phase  with  the  cur- 
rent, is  not  changing,  and  consequently  the  E.M.F.  of  self- 
induction  must  be,  for  the  instant,  zero.  That  is,  /  is  maxi- 
mum when  Eg  is  zero,  which  means  a  displacement  of  90°. 
In  Fig.  30  the  triangle  of  E.M.FSs  of  Fig.  29  is  altered 


Fig.  30. 

to  the  corresponding  parallelogram  of  E.M.F.1?,,  and  the 
maximum  values  substituted  for  the  effective.  If  now  the 
parallelogram  revolve  about  the  center  o,  the  traces  of 
the  harmonic  shadows  of  the  extremities  of  Em,  Erm  and  Esm 
will  develop  as  shown.  It  is  evident  that  the  curve  E/  — 
and  so  also  the  curve  of  current  —  leads  the  curve  E'  by 
the  angle  <p.  It  is  clear  that  the  magnitude  of  <£  depends 
upon  the  relative  values  of  L  and  R  in  the  circuit,  the  exact 
relation  being  derived  from  the  triangle  of  forces. 


tan  0  =  -—  = 

Er 


coL      2  nfL 
RI  =  ~R^      R 


SELF-INDUCTION.  41 

Furthermore 

cos  *  =  f  , 

that  is,  the  cosine  of  the  angle  of  lag  is  equal  to  the  ratio 
of  the  volts  actually  engaged  in  sending  current  to  the 
volts  impressed  on  the  circuit,  and  this  ratio  is  again  equal 
to  the  power-factor  as  stated  in  §  8. 

20.  Instantaneous  Current  produced  by  a  Harmonic 
E.M.F.  in  a  Circuit  having  Resistance  and  Inductance.  — 
The  E.M.F.  at  any  instant  /,  impressed  upon  a  circuit  con- 
taining resistance  and  inductance,  must  be  of  such  magni- 
tude as  to  send  the  instantaneous  current  /'  through  the 
resistance,  and  also  to  neutralize  the  E.M.F.  of  self-induc- 
tion. That  is 

E'  =  RI'  +  L^~- 
dt 

But  E'  =  Em  sin  cot.  (Art.  3.) 

Hence  Em  sin  cat  =  RP  +  L^-', 

dt 

dl'      RT,      Em   . 

~Tt  +  7  1  =  ~f  sm  a)L 
dt       L  L 

f**t 

Multiply  by  integrating  factor  e    L 

f  >v^*.iiiv^*:; 

at  L  L 

/r>  jj  , 

-  dt  =  —  and  writing  in  differential  form, 
J-j  jL 


Rt 


„  p 

dl'e1    +  I'eL      dt  =  ^     sin  cot  .  e 


L 


42  ALTERNATING-CURRENT   MACHINES. 

The  second  term  is  in  the  form  dax  =  ax  loge  a  dx. 

Hence       dl'e1  +  I'd  (eA  =  ^  sin  wt .  e  ^  dt. 

Since  the  first  member  is  in  the  form  d(xy)  =  y  dx  +  x  dy, 

I     -\ 
it  may  be  replaced  by  d(lfeL\.     Integrating, 


Rt 

Fe'1  = 


Rt   T-,         r»  Rt 

Hence 


_£~   I?        f*  _  Kt 

P  =  e   L  -— J  eL  s'mtotdt  + Ce~^  .  (i) 

To  determine  value  of  the  integral,  use  the  formula 

I  u  dv  =  uv  —    Iv  du. 
Let  u  =  sin  tot,  then  du  =  to  cos  tot  dt,  and  let 


=^=|'f!*- 


L  - 

Hence  v  =  -  eL  • 

R 


Then  the  integral  becomes 

//&  r   ^  T    T  Rt 

eLsmtotdt=  --eL  sin  tot  —  —  I  e L  cos  &>/  dt. 
R  R  J 

Use  the  same  formula  again  for  this  second  integral,  but  here 
u  =  cos  wty  hence  du  =  —  to  sin  tot  dt',  and  where 


Rt 

dv  =eL  dt. 


SELF-INDUCTION. 

T     Rt 


43 


Hence 


r-  L  -  cuL2  T 

Then       /  eL  sin  o>/  dt  =   -  ex  sin  cut  -  —e    cos 


^L2 


Ce^s 


Or      i  + 


—  ^  f  eT 
R2  I  J 


sin  ut  dt  =  -  eL  sin  o>/  -  ^--^  cos  o>/. 


Substitute  value  of  integral  in  (i),  then 


-%E 

If  =  ^T 


— 
K 


J2    Rt 

coJu    T 
-—e 
K 


cos  cut 


R2 


R     . 

—  sin  CD  —  a)  cos  cut 

"' 


+  Ce    L. 


Let  the  angle  (j>  be  chosen  so  that  tan  <j>  =  — ,  thus  repre- 
senting the  angle  of  lag  of  the  current  behind  the  E.M.F. 

Therefore 

R 


VR2 


and 


sin  0  = 


cuL 


Hence  R  =  VR2  +  co2L2  cos  6  and  co  = 


VR2  +  cu2L2 

VR2  +  cu2L2  sin  « 


44  ALTERNATING-CURRENT    MACHINES. 

Substitute  these  values  in  the  numerator  of  (2).     Then 


cos  (f>  sin  a>t-R2  +  aj2L2  sin  0  cos  ad"!          - 
2  2 


E  - 

Or     /'  =  —  —  —  (cos  d>  smajt  —  sin  6  coscot)  +  Ce 

+  aSL2 


_ 

The  term  C^  ^  shows  the  natural  rise  of  the  current  when 
the  voltage  is  first  impressed  upon  the  circuit.  After  a  few 
cycles  have  been  completed  this  term  may  be  neglected. 

Then  /'  =          Em     -  sin  (cut  -  (/>).  (4) 


This  expression  gives  the  instantaneous  current  in  a  cir- 
cuit having  resistance  and  inductance  at  any  instant,  when 
a  harmonic  E.M.F.  is  impressed  upon  that  circuit.  When 
L  =  o  and  <£=o,  the  equation  reduces  to 

/'  =  Im  sin  wt  as  in  Article  3. 

21.  Choke  Coils.  —  The  term  choke  coil  is  applied  to 
any  device  designed  to  utilize  counter  electromotive  force 
of  self-induction  to  cut  down  the  flow  of  current  in  an 
alternating-current  circuit.  Disregarding  losses  by  hyster- 
esis, a  choke  coil  does  not  absorb  any  power,  except  that 
which  is  due  to  the  current  passing  through  its  resistance. 
It  can  therefore  be  more  economically  used  than  a  rheostat 
which  would  perform  the  same  functions. 


SELF-INDUCTION.  45 

These  coils  are  often  used  on  alternating-current  circuits 
in  such  places  as  resistances  are  used  on  direct-current 
circuits.  For  instance,  in  the  starting  devices  employed  in 
connection  with  alternating-current  motors,  the  counter 
E.M.F.  of  inductance  is  made  to  cut  down  the  pressure 
applied  at  the  motor  terminals.  The  starter  for  direct- 
current  motors  employs  resistance. 

It  is  often  desirable  to  adjust  the  reactance  of  choke  coils, 
and  for  this  purpose  several  simple  arrangements  may  be 
utilized.  The  coil  may  have  a  sliding  iron  core,  or  its  wind- 
ing may  have  several  taps.  Choke  coils  having  U-shaped 
magnetic  circuits  are  sometimes  provided  with  movable 
polepieces,  which  serve  to  change  the  length  of  the  air  gap. 

Since  a  lightning  discharge  is  oscillatory  in  character  and 
of  enormous  frequency,  a  coil  which  would  offer  a  negligible 
impedance  to  an  ordinary  alternating  current  will  offer  a 
high  impedance  to  a  lightning  discharge.  This  fact  is  recog- 
nized in  the  construction  of  lightning  arresters.  A  choke 
coil  of  but  few  turns  will  offer  so  great  an  impedance  to  a 
lightning  discharge  that  the  high-tension,  high-frequency 
current  will  find  an  easier  path  to  the  ground  through  an 
air  gap  suitably  provided  than  through  the  machinery,  and 
the  latter  is  thus  protected. 

A  choke  coil  for  this  purpose  has  no  iron  core,  and  con- 
sists of  a  few  turns  of  wire,  insulated  from  one  another, 
wound  in  spiral  or  helical  form.  A  lightning  arrester  choke 
coil  used  in  railway  service,  for  station  use,  is  shown  in 

Fig-  3*. 

The  choking  effect  is  not  alone  due  to  the  high  impedance 
offered  to  an  oscillatory  discharge,  but  also  due  to  the  "skin 
effect"  of  the  wire.  By  this  is  meant,  the  tendency  of  the 
alternating  current  to  have  a  greater  density  near  the  surface 


46 


ALTERNATING-CURRENT    MACHINES. 


than  along  the  axis  of  the  conductor,  thus  increasing  the 
resistance.  To  illustrate,  a  ij"  round  copper  conductor 
offers  a  true  resistance,  twice  as  great  as  its  ohmic  resistance, 
to  a  130^  alternating  current.  Even  in  small  wires,  the 
true  resistance  presented  to  currents  of  very  high  frequency, 


Fig.  31 


such  as  those  produced  by  wireless  telegraph  transmitters, 
greatly  exceeds  the  ohmic  resistance,  and  therefore  con- 
ductors, are  required  possessing  a  large  surface  compared 
to  the  cross-section. 

Choke  coils  are  also  used  in  connection  with  alternating- 
current  incandescent  lamps,  to  vary  the  current  passing 
through  them,  and  in  consequence  to  vary  the  brilliancy. 


PROBLEMS.  47 


PROBLEMS. 

1 .  What  is  the  field  winding  inductance  of  a  bi-polar  generator,  having  - 
7500  ampere  turns  per  spool  and  a  total  flux  of  2.4  mega-maxwells, 
when  the  exciting  current  is  2  amperes? 

2.  Find  the  inductance  of  a  cast  steel  test  ring  coil  of  300  turns  when 
carrying  6  amperes,  the  test  ring  being  6"  outside  and  5*  inside  diameter 
and  2\"  in  axial  depth. 

3.  Determine  the  inductance  of  a  pole-line  10  miles  long  and  consist- 
ing of  a  pair  of  No.  i  copper  wires  separated  by  a  distance  between 
centers  of  24  inches. 

4.  Determine  the  self-inductance  of  a  solenoid  consisting  of  10  layers 
of  No.  1 6  double-cotton  covered  wire,  100  turns  per  layer,  wound  upon 
a  cylindrical  wooden  core  2  inches  in  diameter. 

5.  Find  the  value  of  the  current  in  a  circuit  having  5  ohms  resistance 
and  an  inductance  of  0.15  henry,  .03  seconds  after  impressing  no  volts 
upon  that  circuit. 

6.  What  is  the  time  constant  of  a  circuit  in  which  the  current  reaches 
half  of  its  ultimate  value  .0018  second  after  connection  with  a  source  of 
E.M.F.  ? 

7.  What  would  be  the  current  .02  second  after  suppressing  the  E.M.F. 
in  the  circuit  of  problem  5,  a  constant  flow  having  been  previously 
established  ? 

8.  Determine  the  energy  stored  in  the  magnetic  field  of  the  generator 
of  problem  i,  assuming  L  to  be  constant  during  the  rise  or  fall  of  the 
current. 

9.  Find  the  current  produced  by  a  60  ~~  alternating  E.M.F.  of  120 
volts  in  a  circuit  having  10  ohms  resistance  and  an  inductance  of  .04 
henry.     What  is  the  power  factor  of  the  circuit  ? 

10.  What  should  the  inductance  of  the  circuit  of  problem  9  be,  to 
attain  a  power  factor  of  85%? 

11.  Derive  an  expression  for  the  current  in  a  circuit  whose  resistance 
and  reactance  are  equal.     What  will  be  the  power  factor? 

12.  Find    the  instantaneous  value    of  a   25-^  alternating  current, 
2.342  seconds  after  impressing  a  harmonic  E.M.F.  of  125  volts  maxi- 
mum upon  a  circuit  which  has  a  resistance  of  8  ohms  and  an  induct- 
ance of  0.04  henry. 


48  ALTERNATING-CURRENT  MACHINES. 


CHAPTER    III. 

CAPACITY. 

22.  Condensers.  —  Any  two  conductors  separated  by  a 
dielectric  constitute  a  condenser.  In  practice  the  word  is 
generally  applied  to  a  collection  of  thin  sheets  of  metal 
separated  by  thin  sheets  of  dielectric,  every  alternate  metal 
plate  being  connected  to  one  terminal  and  the  intervening 
plates  to  the  other  terminal.  The  Leyden  jar  is  also  a 
common  form  of  condenser. 

The  function  of  a  condenser  is  to  store  electrical  energy 
by  utilizing  the  principle  of  electrostatic  induction.  When- 
ever a  difference  of  potential  is  impressed  upon  the  con- 
denser terminals,  stresses  are  set  up  in  the  dielectric  which 
exhibit  themselves  electrically  as  a  counter  electromotive 
force,  opposing  and  neutralizing  the  impressed  E.M.F. 
During  the  period  of  establishment  of  the  stresses  a  current 
flows  through  the  dielectric,  and  it  is  known  as  a  displace- 
ment current.  This,  however,  ceases  to  flow  as  soon  as 
the  counter  electromotive  force  of  dielectric  polarization  is 
equal  in  magnitude  to  the  impressed  E.M.F.  The  con- 
denser is  then  said  to  be  charged.  It  should  be  remem- 
bered that  the  charge  resides  in  the  dielectric  as  the  result 
of  the  stresses  produced  in  it  by  the  impressed  E.M.F. 

The  nature  of  the  stresses  in  the  dielectric  can  be  more 
readily  understood  by  considering  the  conductors  to  be 
surrounded  by  an  electrostatic  field.  This  field  may  be 
considered  as  composed  of  electrostatic  lines  of  force,  shown 


CAPACITY.  49 

in  Fig.  32,  which  indicate  by  their  directions  the  directions 
of  the  stresses,  and  by  their  nearness  to  each  other  the 
magnitude  of  the  stresses.  The  greater  the  impressed 
E.M.F.,  the  greater  will  be  the  number  of  these  lines  and 


the  greater  will  be  the  charge.  The  property  of  a  dielectric 
which  opposes  the  passage  of  this  dielectric  flux  may  be 
termed  its  obstructance,  and  it  is  similar  in  this  respect  to 
reluctance  in  opposing  the  passage  of  magnetic  flux,  and  to 
resistance  in  opposing  the  flow  of  current.  The  obstruc- 
tivity  of  a  dielectric  is  three  hundred  times  the  reciprocal 
of  its  specific  inductive  capacity  or  its  dielectric  constant, 
which  is  the  ratio  of  the  electric  strain  to  the  stresses  pro- 
duced by  it  in  the  dielectric. 

No  dielectric  is  capable  of  supplying  more  than  a  definite 
maximum  amount  of  counter  electromotive  force  per  unit 
of  length  measured  along  a  line  of  force.  If  the  impressed 
potential  difference  exceeds  this  maximum  counter  E.M.F., 
which  is  the  measure  of  its  dielectric  strength,  the  dielectric 
is  ruptured  and  breaks  down  mechanically.  Of  course,  if 


ALTERNATING-CURRENT    MACHINES. 


the  dielectric  is  a  liquid  or  a  gas,  it. will  be  restored  to  its 
original  state  when  the  impressed  E.M.F.  is  diminished. 
At  the  point  of  rupture,  a  current  in  the  form  of  a  spark  or 
an  arc  passes  from  one  conductor  to  the  other,  the  tendency 
being  to  lessen  the  potential  difference.  Such  rupture, 
followed  by  an  arc,  is  a  frequent  source  of  trouble  in  electric 
machinery. 

The  dielectric  strength  of  any  dielectric  depends  upon 
its  thickness,  the  form  of  the  opposed  conducting  surfaces, 
and  the  manner  in  which  the  E.M.F.  is  applied,  whether 
gradually,  suddenly,  or  periodically  varying.  It  has  been 
stated  that  the  dielectric  strength  approximately  varies 
inversely  as  the  cube  root  of  the  thickness,  showing  that  a 
thin  sheet  is  relatively  stronger  than  a  thick  one  of  the  same 
material.  For  example,  the  dielectric  strength  of  crystal 
glass  when  5  mm.  thick  is  183  kilo  volts  per  centimeter,  but 
when  i  mm.  thick  it  is  285  kilovolts  per  centimeter.  In  the 
following  table  giving  the  dielectric  strengths  of  various 
materials,  the  particular  thicknesses  for  which  the  values 
are  given  are  stated: 


Material. 

Thickness  in  mm. 

Dielectric  Strength 
in  Kilovolts  per  cm. 

Air 

10 

29.8' 

Air 

I 

43-6 

Glass 

5 

l83 

Mica 

i 

610 

Mica 

O.I 

1150 

Micanite 

i 

400 

Linseed  Oil 

6 

84 

Vaseline  Oil 

6 

60 

Lubricating  Oil 

6 

48 

Ebonite 

2 

43° 

The  capacity  of  a  condenser  is  numerically  equal  to  the 
quantity  of  electricity  with  which  it  must  be  charged  in 


CAPACITY.  51 

order  to  raise  the  potential  difference  between  its  terminals 
from  zero  to  unity. 

If  the  quantity  and  potential  be  measured  in  c.  G.  s. 
units,  the  capacity,  c,  will  be  in  c.  G.  s.  units.  If  practical 
units  be  employed,  the  capacity,  c,  is  expressed  in  jarads. 
The  farad  is  the  practical  unit  of  capacity.  A  condenser 
whose  potential  is  raised  one  volt  by  a  charge  of  one 
coulomb  has  one  farad  capacity.  The  farad  is  io~9  times 
the  absolute  unit,  and  even  then  is  too  large  to  conven- 
iently express  the  magnitudes  encountered  in  practice. 
The  term  microfarad  (TOTTUUO*)  farad)  is  in  most  general 
use. 

In  electrostatics,  both  air  and  glass  are  used  as  dielec- 
trics in  condensers;  but  the  mechanical  difficulties  of  con- 
struction necessitate  a  low  capacity  per  unit  volume,  and 
therefore  render  these  substances  impracticable  in  electro- 
dynamic  engineering.  Mica,  although  it  is  expensive  and 
difficult  of  manipulation,  is  generally  used  as  the  dielectric 
in  standard  condensers  and  in  those  which  are  intended 
to  withstand  high  voltages.  Many  commercial  condensers 
are  made  from  sheets  of  tinfoil,  alternating  with  slightly 
larger  sheets  of  paraffined  paper.  Though  not  so  good  as 
mica,  paraffin  will  make  a  good  dielectric  if  properly 
treated.  It  is  essential  that  all  the  moisture  be  expelled 
from  the  paraffin  when  employed  in  a  condenser.  If  it 
is  not,  the  water  particles  are  alternately  attracted  and 
repelled  by  the  changes  of  potential  on  the  contiguous 
plates,  till,  by  a  purely  mechanical  action,  a  hole  is  worn 
completely  through  the  dielectric,  and  the  whole  condenser 
rendered  useless  by  short-circuit.  Ordinary  paper  almost 
invariably  contains  small  particles  of  metal,  which  become 
detached  from  the  calendar  rolls  used  in  manufacture. 


52  ALTERNATING-CURRENT    MACHINES. 

These    occasion    short-circuits    even    when    the    paper    is 
doubled. 

The  capacity  of  a  condenser  is  proportional  directly  to 
the  area  and  inversely  to  the  thickness  of  the  dielectric. 
It  is  also  directly  proportional  to  the  dielectric  constant  of 
the  insulating  material,  which,  in  addition  to  the  definition 
already  cited,  may  be  defined  as  the  number  expressing  the 
ratio  of  increase  of  the  capacity  of  an  air  condenser,  when 
the  air  is  entirely  replaced  by  that  dielectric.  This  constant, 
usually  represented  by  K,  decreases  with  the  temperature 
and  with  the  time  of  charge.  For  these  reasons  the  values 
of  K  given  by  different  observers  differ  considerably,  but 
some  accepted  values  are  given  in  the  following  table: 

DIELECTRIC    CONSTANTS    AT    15°  C. 


Flint  Glass  (dense)  10.1 

Flint  Glass  (light)  6.57 

Crown  Glass  (hard)  6.96 

Mica  6.64 

Tourmaline  6.05 


Quartz  4.55 

Sulphur  2.9    to  4.0 

•Shellac  2.7    to  3.0 

Ebonite  2.05  to  3.15 

Paraffin  Wax  2.0    to  2.3 


The  resistance  of  a  condenser  is  not  infinite,  but  a  meas- 
urable quantity,  and  is  usually  expressed  in  megohms  per 
microfarad,  or,  when  referring  to  cables,  in  megohms  per 
mile.  Hence  there  is  always  a  leakage  from  one  charged 
plate  to  the  other,  both  through  the  dielectric  and  over  its 
surface.  Poor  insulation  may  occasion  a  considerable  loss 
of  energy  appearing  in  the  form  of  heat,  and  is  therefore 
to  be  avoided. 

Analogous  to  magnetic  hysteresis  in  iron,  is  dielectric 
hysteresis  in  condensers,  but,  contrary  to  the  former,  it 
decreases  as  the  frequency  increases.  Thus,  at  a  frequency 
of  the  order  of  10  million  cycles,  dielectric  hysteresis  is 
entirely  absent.  A  dielectric  having  a  high  hysteretic  con- 


CAPACITY.  53 

stant,  such  as  glass  —  6.1,  may  consume  a  considerable 
amount  of  energy  on  low  frequency  circuits,  this  loss  also 
appearing  as  heat. 

23.  Capacity  Formulae.  —  The  following  formulae,  in 
which  r  is  the  radius  of  the  conductor  and  /  its  length,  both 
in  centimeters,  give  the  capacity  in  microfarads  of  con- 
ductors with  respect  to  the  earth: 

Sphere  in  free  space, 


900,000 

Circular  disk  in  free  space, 
r 


1,413,72° 

One  cylindrical  wire  in  free  space, 
/-•  _ ' 


4,144,680  Iog10  - 

One  cylindrical  wire  h  cm.  from  the  earth, 

c==   / 

4,144,680  Iog10  y-  • 

In  the  following  formulae,  giving  the  capacity  of  con- 
densers of  various  forms,  only  that  portion  of  the  dielectric 
flux  which  passes  perpendicularly  between  the  conducting 
surfaces  is  considered;  that  is,  the  end  flux  shown  by  the 
curved  dotted  lines  in  Fig.  32  is  neglected.  Under  this 
consideration,  the  following  expressions  may  only  be  used 
when  the  thickness  of  the  dielectric  is  very  small  compared 
to  the  conductor  area. 


54  ALTERNATING-CURRENT    MACHINES. 


Two  concentric  spheres, 
n  r  r  K 


. 
where  r,  >  r.. 


900,000  (fj-n) 
Two  concentric  cylinders, 

IK 

C  = -  where  ; 

4,144,680  Iog10  -2 

Two  cylindrical  wires  d  cm.  apart, 
IK 


8,289,360  Iog10  - 
Two  circular  plates,  d  cm.  apart, 


_ 

3,600,000  d 

From  this  last  formula,  another  may  be  readily  derived 
for  the  calculation  of  the  capacity  of  a  condenser  having  n 
dielectric  sheets,  and  having  its  symbols  expressed  in  inches. 
The  capacity  is 

/-  An  T, 

C  =  .000225  —  ^ 
t 

where  A  is  the  area  of  each  sheet  in  square  inches,  and  t  is 
its  thickness  in  mils. 

The  following  data  of  a  condenser,  used  in  duplex 
telegraphy,  give  an  idea  of  capacity  and  dielectric  resistance. 

The  condenser  consists  of  tinfoil  and  paper  sheets,  the 
former  being  brought  out  alternately  to  one  terminal  and 
then  to  the  other.  There  are  92  sheets  of  beeswaxed  paper, 
7X5  inches  and  two  mils  thick,  which  constitute  the  dielec- 
tric. The  capacity  of  the  condenser  is  1.47  microfarads, 
and  its  dielectric  resistance  is  160  megohms. 


CAPACITY. 


55 


24.  Connection  of  Condensers  in  Parallel  and  in  Series. 
—  Condensers  may  be  connected  in  parallel  as  in  Fig.  33. 
If  the  capacities  of  the  individual 
condensers  be  respectively  C,,  C2,  C3, 
etc.,  the  capacity  C  of  the  combina- 
tion will  be  — 

C  =  C,  +  C9  +  C,  +  . 


Fig.  33- 


For  the  potential  difference  on  each 

condenser  is  the  same,  and  equal  to 

the  impressed  E.M.F.,  and  the  total  charge  is  equal  to  the 

sum  of  the  individual  charges,  or 

E  =  E,  =  E2  =  E3  =  .  .  .  . 
and  Q  =  Ql  +  Q2  +  Q3  +  .  .  .  . 


Then  by  division  —  =  -^ 
E      Et 

But  by  definition  -  =  C, 
E 


+  &+£+..., 

A2  ^3 

^-  =  Cj  and  so  on, 


therefore 


C  =  C,  +  C2  +  C3  + 


The  parallel  arrangement  of  several  condensers  is  equiva- 
lent to  increasing  the  number  of  plates  in  one  condenser. 
An  increase  in  the  number  of  plates  results  in  an  increase  in 
the  quantity  of  electricity  necessary  to  raise  the  potential 
difference  between  the  terminals  of  the  condenser  one  volt; 
that  is,  an  increase  in  the  capacity  results. 

If  the  condensers  be  connected  in  series,  as  in  Fig.  34, 
the  capacity  of  the  combination  will  be 


$6  ALTERNATING-CURRENT   MACHINES. 

For,  if  a  quantity  of  positive  electricity,  Q,  flow  into  the 
left  side  of  C^  it  will  induce  and  keep  bound  an  equal  neg- 
ative quantity  on  the  right  side  of  Cl9  and  will  repel  an 
equal  positive  quantity.  This  last  quantity  will  constitute 
the  charge  for  the 
left  side  of  C2. 
The  operation  is 

«  ----  Er  ----  >r<  ---  -Ej,  ------  .*•<•  ---  E-3-  ----  W 

repeated     in    the         ,U  -----------------  B—  —  »4 

case    of    each    of  Fig-  34> 

the  condensers.  It  is  thus  clear  that  the  quantity  of 
charge  in  each  condenser  is  Q.  The  impressed  E.M.F. 
must  consist  of  the  sum  of  the  potential  differences  on  the 
separate  condensers.  Let  these  differences  be  respectively 
E^  Ey  E^  etc.  Then  the  impressed  E.M.F. 

E  =  E^  +  £2  +  £s  +  -  •  •  • 
But  ^  =  £        ^2  =  7^        ^3=-i>    etc., 

C'l  C2  Cg 

and  also,  £  =  —, 


therefore  =       +      = 

C  C-!          C2          C-3 


As  an  example,  consider  three  condensers  of  respective 
capacities  of  i,  2,  and  5  microfarads.  Since  the  factor  to 
reduce  to  farads  will  appear  on  both  sides  of  the  equations, 
it  may  here  be  omitted.  With  the  three  in  multiple  (Fig. 
33),  the  capacity  of  the  combination  will  be 

C  =  i  +  2  -f  5  =  8  mf . 


CAPACITY.  57 

With  the  three  in  series  (Fig.  34), 

C  = =  .588  mf. 

1       I       1 
i       2       5 

With  the  two  smaller  in  parallel  and  in  series  with  the 
larger  (Fig.  35), 

C  = I =  1.875  mf. 


r+i  +  5 


Fig.  35- 


With  the  two  smaller  in  series  and  in  parallel  with  the 
larger  (Fig.  36), 


C= 


I  2 


=  5-666 


If  with  any  condensers 

Cl=  C2=  C3=  ----  =Cn, 

then,  with  n  in  multiple, 


and  with  n  in  series, 


C  =  -  Ci. 
n 


It  is  interesting  to  note  that  the  formulas  for  capacities 
in  parallel  and  in  series  respectively  are  just  the  reverse  of 
those  for  resistances  in  parallel  and  in  series  respectively. 

25.   Decay  of  Current  in  a   Condensive  Circuit.  —  The 

opposition  to  a  flow  of  current  which  is  caused  by  a  con- 


5&  ALTERNATING-CURRENT   MACHINES. 

denser  is  quite  different  -from  that  which  is  caused  by  a 
resistance.  To  be  sure,  there  is  some  resistance  in  the 
leads  and  condenser  plates,  but  this  is  generally  so  small 
as  to  be  negligible.  The  practically  infinite  resistance  of 
the  condenser  dielectric  does  not  obstruct  the  current  as 
an  ordinary  resistance  is  generally  considered  to  do.  The 
dielectric  is  the  seat  of  a  polarization  E.M.F.  which  is  de- 
veloped by  the  condenser  charge  and  which  grows  with  it. 
It  is  a  counter  E.M.F. ;  and  when  it  reaches  a  value  equal 
to  that  of  the  impressed  voltage,  the  charging  current  is 
forced  to  cease. 

To  find  the  current  at  any  instant  of  time,  /,  in  a  circuit 
(Fig.  37)  containing  a  resistance  R  and  a  capacity  C,  the 
constant  impressed  pressure  E  must 
be  considered  as  consisting  of  two 
variable  parts,  one  Er,  being  active 
in   sending  current  through  the  re- 
sistance, and    the    other    part,  Ec, 
being   required   to    balance  the   po- 
tential of  the  condenser.     Then  at 
all  times 

Let  time  be  reckoned  from  the  instant  the  pressure  E  is 
applied ;  when,  therefore,  t  =  o  and  70  =  —.     Consider  the 

current  at  any  instant  of  time  to  be  /'.  Then  if  it  flow 
for  dt  seconds  it  will  cause  dQ  coulombs  to  traverse  the 
circuit,  and 

f'=  d-w or 

from  which 


<r=//v, 


CAPACITY. 


59 


c- 

c- 


By  definition, 

therefore, 

2 

And  by  Ohm's  Law, 

Er'  =  /'  R, 

so  at  this  instant  of  time 


C 


E'  =  £,.' 


whence 


E'C^  RCI'  +  Cl'dt, 

which  upon  differentiating,  becomes 

o  =  RCdl'  +  I'dt. 
Integrating  C* #  =  _  RC  C*  ^9 

Jo  J/o       / 


CONDENSER 
CHARGING  CURRENT 
E-100V. 
R-10 
C  — 2  MF. 
_.  000002  F. 


Solving  for  /', 


which      is      the      expression 
sought.        Like     the     corre- 
sponding   expression    for   an 
{  inductive    circuit,    it   is  loga- 

rithmic. 

Fig.  38  is  a  curve  showing  the  decay  of  current  in  a 
condenser  for  the  conditions  indicated.     The  product  RC 


60  ALTERNATING-CURRENT   MACHINES. 

is  the  time  constant  of  a  condensive  circuit  and  is  similar 
to  the  ratio  —  in  an  inductive  circuit. 

K 

26.  Energy  Stored  in  Dielectric.  —  A  current  7  flowing 
in  a  condensive  circuit  against  a  dielectric  polarization  of 
E  volts,  represents  a  power  of  El  watts.  The  work  per- 
formed in  an  interval  of  time  dt  is 

dW  =  EJdt 

and  represents  the  elementary  work  done  in  establishing 
the  stresses  in  the  dielectric.     Since 

E  =    2  and  I  dt  =  dQ, 
O 

there  results  by  substitution 


which  when  integrated  through  the  full  range,  that  is 

/w          T     rq 
dW  =  -^      QdQ, 
CJo 

O2 
becomes  W  =    -~r  joules. 

2  Cx 

This  is  the  expression  for  the  energy  required  to 
establish  the  dielectric  stresses  when  the  current  is  first 
applied,  and  also  the  expression  for  the  energy  returned 
to  the  circuit  by  the  dielectric  when  the  impressed  E.M.F. 
is  withdrawn. 

27.  Condensers  in  Alternating-Current  Circuits  —  Hy- 
draulic Analogy.  —  Imagine  a  circuit  consisting  of  a  pipe 
through  which  water  is  made  to  flow,  first  one  way,  then 


CAPACITY.  6l 

the  other,  by  a  piston  oscillated  pump-like  in  one  section 
of  it.  The  pipe  circuit  corresponds  to  an  electric  circuit, 
the  pump  to  a  generator  of  alternating  E.M.F.,  and  the 
flow  of  water  to  a  flow  of  alternating  current.  Further 
imagine  one  section  of  the  pipe  to  be  enlarged,  and  in  it 
placed  a  transverse  elastic  diaphragm.  This  section  cor- 
responds to  a  condenser.  Its  capacity  with  a  unit  pressure 
of  water  on  one  side  depends  upon  the  area  of  the  dia- 
phragm, its  thinness,  and  the  elastic  coefficients  of  the 
material  of  which  it  is  made.  In  a  condenser  the  capacity 
depends  upon  the  area  of  the  dielectric  under  strain,  its 
thinness,  and  the  specific  inductive  capacity  of  the  dielec- 
tric employed.  As  the  water  surges  to  and  fro  in  the 
pipe,  some  work  must  be  done  upon  the  diaphragm,  since 
it  is  not  perfectly  elastic.  This  loss  corresponds  to  the 
loss  in  a  condenser  by  dielectric  hysteresis.  The  fact  that 
the  diaphragm  is  not  absolutely  impervious  to  water  cor- 
responds to  the  fact  that  a  dielectric  is  not  an  absolute 
electric  insulator.  As  the  diaphragm  may  be  burst  by  too 
great  a  hydrostatic  pressure,  so  may  the  dielectric  be  rup- 
tured by  too  great  an  electric  pressure. 

28.  Phase  Relations.  —  To  understand  the  relation  be- 
tween pressure  and  current  in  a  condensive  circuit,  con- 
sider the  above  analogy.  Imagine  the  diaphragm  in  its 
medial  position,  with  equal  volumes  of  water  on  either  side 
of  it,  and  the  piston  in  the  middle  of  its  travel.  This 
middle  point  corresponds  to  zero  pressure.  When  the  pis- 
ton is  completely  depressed  there  is  a  maximum  negative 
pressure,  when  completely  elevated,  a  maximum  positive 
pressure,  if  pressure  and  flow  upward  be  considered  in  the 
positive  direction.  If  the  piston  oscillate  in  its  path  with  a 


62 


ALTERNATING-CURRENT   MACHINES. 


regular  motion,  it  is  clear  that  the  water  will  flow  upward 
from  the  extreme  lowest  to  the  extreme  highest  position  of 
the  piston.  That  is,  there  will  be  flow  in  the  positive  direc- 
tion from  the  maximum  negative  to  the  maximum  positive 
values  of  pressure.  The  direction  of 
flow  is  seen  to  remain  unchanged  while 
the  piston  passes  through  its  middle 
position  or  the  point  of  zero  pressure. 
These  facts  are  indicated  in  Fig.  39, 
o  FLOW  which  shows  that  portion  of  the  pipe 
having  the  piston  and  the  diaphragm. 

Returning  to  electric  phenomena,  if  a 
harmonic  E.M.F.  be  impressed  upon 
any  circuit,  a  harmonic  current  will  flow 
in  it.  So  in  a  circuit  containing  a  con- 
denser and  subject  to  a  sinusoidal 
E.M.F. ,  the  current  flow  will  be  sinusoidal.  This  flow 
will  be  in  the  positive  direction  from  the  negative  maximum 
to  the  positive  maximum  of  pressure,  and  in  a  negative 
direction  from  the  positive 
maximum  to  the  negative 
maximum,  as  described 
above.  This  necessitates 


Fig.  40. 


Fig.  39- 


that  the  zero  values  of 
current  occur  at  the  maxi- 
mum values  of  pressure; 
and  since  the  curves  are  both  sinusoids,  their  relation 
may  be  plotted  as  in  Fig.  40.  It  is  immediately  seen  that 
these  curves  are  at  right  angles,  as  described  in  §  7,  and  that 
the  current  leads  the  pressure  by  90°. 

Reference  again  to  the  hydraulic  analogy  will  show  that 
the   condenser  is   completely   charged   at   the   instant   of 


CAPACITY.  63 

maximum  positive  pressure,  discharged  at   the  instant  of 
zero  pressure,   charged  in  the  opposite  direction   at  the 
instant  of  maximum  nega- 
tive  pressure,  and  finally 
discharged  at  the   instant 
of  the  next  zero  pressure. 
Thus   the   charge   is  zero 
at  the  maximum  current 
flow,  and  at  a  maximum 
at  zero    current,    that    is, 
when    the    current    turns 
and    starts   to    flow   out. 


Fig.  41. 

These   points   are   marked   in 


Fig.  41. 


29.   Current    and   Voltage    Relations.  —  If   a   sinusoidal 
pressure  E  of  frequency  /  be  impressed  upon  a  condenser, 

the  latter  is  charged  in  — -  seconds,  discharged  in  the  next 
4  / 

—  seconds,  and  charged  and  discharged  in  the  opposite 

4  / 

direction  in  the  equal  succeeding  intervals.     The  maximum 

voltage  Em  =  ^2  E  (§  5),  hence  the  quantity  at  full  charge 

is 

Qm  =  V7  EC. 

The    quantity  flowing  through    the  circuit    per  second  is 

4/<2™  =  4/v/2EC. 
This  number  therefore  represents  the  average  current,  or 

Tav  =  4  V2JEC. 
From  §  5,  the  effective  current 


[  \/2 


64 


ALTERNATING-CURRENT   MACHINES. 


7  =  2  7T/CE, 


E  = 


-I. 


whence 

and  ~  - 

2  7T/C 

The  last  is  an  expression  for  the  volts  necessary  to  send 
the  capacity  current  through  a  circuit.  The  expression 
1/2  TT/C  is  called  the  capacity  reactance  of  the  circuit.  It  is 
analogous  to  2  TtfL,  the  inductive  reactance  of  an  inductive 
circuit. 

If  the  circuit  contain  both  a  resistance  R  and  a  capa- 
city C,  the  voltage  E  impressed  upon  it  must  be  considered 
as  made  up  of  two  parts,  Erj  which  sends  current  through 
the  resistance  and  is  therefore  in  phase  with  the  current, 
and  Ec,  which  balances  the  counter  pressure  of  the  con- 
denser and  is  therefore  90°  behind  the  current  in  phase. 

By  Ohm's  Law 

Er  =  .RI, 
and  from  above 


The  impressed  E  must  overcome  the  resultant  of  these 
two  E.M.-F.'s;  and  since  they  are  at  right  angles 

E  =  \/Er    +  E?. 


T     __ 


E 


Fig.  42. 


The  relation  of  the  E.M.F.'s  is  shown  graphically  in 
Fig.  42,  where  the  current,  which  is  in  phase  with  the 
pressure  Er,  is  seen  to  lead  the  impressed  pressure  by  the 
angle  <£. 


CAPACITY.  65 

30.  Instantaneous  Current  in  a  Circuit  Having  Capacity 
and  Resistance.  —  The  value  of  the  E.M.F.,  impressed 
upon  a  circuit  containing  capacity  and  resistance  at  any 
instant  t,  must  be  sufficient  to  send  the  instantaneous  cur- 
rent /'  through  the  resistance  and  also  neutralize  the  E.M.F. 
of  dielectric  polarization.  Hence 

But  E'  =  Em  sin  cot,  (§3) 

£/  =  I'R,  (§  25) 

and  £/  =  J  rd*  (§  25) 

C 

therefore                   Em  sin  ait  = 
Differentiating  cuEm  cos  a>t  dt  =  R  dP  + < 


fl'dt 

PR  +  -  -  • 


__ 

Multiplying  by  integrating  factor  e^  RC  or  eRC  and  dividing 
by  R,  this  becomes 


dl'e«c  +  Ir'-~c  =  ~-e       cos  ut  dt. 

The  second  term  is  in  the  form  dax  =  ax  \oge  a  dx,  hence 
this  is 


e?  cos  wt  dL 


Since  the  first  member  is  in  the  form  d(xy)  =  y  dx  +  x  dy  it 

I     —\ 
equals  d  \reRC).     Substitute  and  integrate,  then 

t  p       p    t 

peRC    =    _Jp-  J   6RC  CQS  wt  dt    +    Cf 


66 


or 


ALTERNATING-CURRENT   MACHINES. 


R 


•/«*' 


RC  cos  o)t  dt  +  Ce 


(i) 


To    determine    value    of    the    integral,    use    the    formula 

I  u  dv  —  uv  —  Iv  du,  where  u  =  cos  cot,  hence  du  =  —  co 

sin  cot  dt\  and  where  dv  =  eRC  dt  =  RCeRC-^r,  hence  v  = 

RCe1^.     Then 

/t  t  r*    t 

eRC  cos  cot  dt  =  RCeRC  cos  cot  +  RCco  J  eRC  sin  cot  dt. 

The  second  integral  is  in  the  same  form,  but  here  u  =  sin 
cot,  hence  du  =  co  cos  cot  dt,  dv  and  v  remain  the  same.    Then 

/t  t  t 

eRO  cos  cot  dt  =  RCeRC  cos  cot  +  R2C2coeRC'sm  cot  -R2C2co2 
/_  t 
eRC  cos  cot  dt. 

(i  +  R2C2aA  j  e™ cos  cot  dt  =  RCe1^  cos  cot 

t 
+  R2C2coeRC<  sin  cot. 

Substitute  value  of  integral  in  (i),  there  results 

[/  t  i 
RCeRC    cos  cot  -f  R2C2coeRC  sin  cot 
4-  R2C2co2 
•                                       J 

[cos  cot 
I 


+  Ce    RC, 

r  =  COCE, 


+  RCco  sin  a), 
~R2C2co2 


r  -  Eni 


—  cos  cot  -}-  R  sin  cot 


™  +  *2 


Ce 


CAPACITY.  67 

Let  the  angle  <£  be  chosen  so  that 
i 

tan  0  =  °£  =  -!-  , 
£       ^C^ 

thus  representing  the  angle  by  which  the  current  leads  the 
E.M.F.     Therefore 


and  — 

By  substitution  in  (2),  there  results, 

vfcisEi  .  -i- 

/'  =  £m [sin  <f>  cos  a>t  +  cos  ^>  sin  cut]  +  Ce    RC , 


where  the  exponential  term  shows  the  natural  current  decay 
in  a  condensive  circuit  when  the  E.M.F.  is  first  applied. 
Neglecting  this  term,  (3)  reduces  to 

1'  -      ,—  ^==  sin  (ut  +  ^),  (4) 


giving  an  expression  for  the  instantaneous  current  in  a 
circuit,  having  resistance  and  capacity,  at  any  instant  when 
a  harmonic  E.M.F.  is  impressed  upon  that  circuit.  When 
the  capacity  of  a  circuit  is  an  infinitesimal,  such  as  is  the  case 
when  its  two  terminals  are  slightly  separated,  then  in  the 


68  ALTERNATING-CURRENT   MACHINES. 

formula,  C  =  o  and  the  current  is  also  zero,  which  is 
evidently  true  for  an  open  circuit.  When  the  circuit  con- 
tains no  capacity  relative  to  itself,  and  only  resistance,  then 

/  i  \2 

the  term  1—7,1  should  not  enter  the  equation,  which  will 
\ajC/ 

then  reduce  to 

P  =  Im  sin  (cot  +$),         as  in  §  3. 

PROBLEMS. 

1.  Determine  the  capacity  of  a  pair  of  No.  ooo  line  wires,  two  feet 
apart,  and  three  miles  long! 

2.  Calculate  the  dielectric  constant  of  the  condenser  mentioned  in 
§  23.     What   is   its   insulation   resistance   expressed   in   megohms   per 
microfarad  ? 

3.  Derive  the  formula  C  =  .000225  —  K  of  §  23. 

» 

4.  Find  the  equivalent  capacity  of  the  group  of  condensers  shown  in 


Fig.  43. 

Fig.  43,  the  number  adjacent  to  each  condenser  representing  its  capacity 
in  microfarads. 

5.  If  a  constant  E.M.F.  of  150  volts  is  applied  to  the  terminals  A 
and  B  of  the  group  of  condensers  shown  in  Fig.  43,  what  will  be  the 
voltage  across  the  terminals  of  each  condenser? 

6.  If  a  circuit  having  a  resistance  of  10  ohms  and  a  capacity  of  20 
microfarads  has  a  constant  E.M.F.  of  100  volts  impressed  upon  it,  how 
long  will  it  take  for  the  current  to  sink  to  half  its  initial  value? 

7.  Determine  the  energy  which  can  be  electrically  stored  in  a  cubic 
inch  of  mica  dielectric  when  the  applied  potential  is  450  volts  per  mil 
thickness. 


PROBLEMS.  69 

8.  Find  the  current  produced  by  a  25^-  alternating  E.M.F.  of  100 
volts  in  a  circuit  having  25  ohms  resistance  and  a  capacity  of  30  micro- 
farads.    What  is  the  power  factor  of  the  circuit  ? 

9.  It  is  desired  to  construct  a  condenser  of  crown  glass  plates  10  X  12 
inches  so  that  the  power  factor  of  its  circuit  having  12.5  ohms  resistance 
shall  be  90%  for  an  oscillatory  current  of  80,000  cycles.     How  many 
plates  will  be  required  if  the  thickness  of  each  is  .15  inch? 

10.  Determine  the  instantaneous  value  of  a  60*—'  alternating  current 
5.71  seconds  after  impressing  a  harmonic  E.M.F.  of  220  volts  (effective) 
upon  a  circuit  having  a  resistance  of  100  ohms  and  a  capacity  of  25 
microfarads. 


ALTERNATING-CURRENT    MACHINES. 


CHAPTER   IV. 

ALTERNATING-CURRENT  CIRCUITS. 

31.  Resistance,  Inductance  and  Capacity  in  an  Alter- 
nating-Current Circuit.  —  In  general,  alternating-current 
circuits  have  resistance,  inductance  and  capacity.  An 
expression  for  the  current  flow  in  such  a  circuit  may  be 
derived  mathematically,  as  in  §  34,  or  the  current  may  be 
found  graphically  by  combining  results  already  obtained. 
In  §  19  it  was  shown  that  the  counter  E.M.F.  due  to  the 
inductive  reactance  of  a  circuit  is  2  xjLI  and  leads  the 
current  by  90°,  and  in  §  29  it  was  shown  that  the  E.M.F. 


V 


Fig-  45- 

of  dielectric  polarization  due  to  the  capacity  reactance  of  a 
circuit  is and  lags  behind  the  current  by  90°;  hence 

2  TtjC 

these  two  E.M.F. ,'s  are  opposite  in  phase,  or  180°  apart. 
These  relations  are  shown  in  Fig.  44,  where  the  inductive 
reactance  is  greater  than  that  due  to  capacity,  and  in  Fig. 
45,  where  the  latter  exceeds  the  former,  the  resistance  being 
the  same  in  both  cases.  The  common  factor  I  is  omitted 


ALTERNATING-CURRENT   CIRCUITS.  71 

in  these  diagrams,  as  is  very  often  done  for  convenience, 
but  it  should  be  remembered  that  neither  resistance,  reac- 
tance nor  impedance  is  a  vector  quantity.  Clearly  the 
impedance  resulting  from  the  three  factors,  R,  L  and  C,  is 
represented  in  direction  and  in  magnitude  by  the  hypothe- 
nuse  as  shown,  and  the  impressed  pressure  is  I  times  this 
quantity. 

The  general  expression  for  the  flow  of  an  alternating 
current  through  any  kind  of  a  circuit  is  therefore 

E 


2T/CJ 

the  quantity  within  the  brackets  indicating  an  angle  of  lag 
of  current  when  positive,  and  an  angle  of  lead  when  negative. 

32.  Definitions  of  Terms.  —  In  considering  the  flow  of 
alternating  currents  through  series  circuits  and  through 
parallel  circuits,  continual  use  must  be  made  of  various 
expressions,  some  of  which  have  been  defined  during  the 
development  of  the  previous  chapters.  For  convenience 
the  names  of  all  the  expressions  connected  with  the  general 
equation 

E 


r  __ 


\/R*  +  (2  7T/L    - 


27T/C/ 

will  be  given  and  denned. 

I  is  the  current  flowing  in  the  circuit.  It  is  expressed 
in  amperes,  and  lags  behind  or  leads  the  pressure,  by  an 
angle  whose  value  is 

2^L  ~~—Jr 

.  2  7T/C 

<£  =  tan'1 *—  . 


72  ALTERNATING-CURRENT    MACHINES. 

E  is  the  harmonic  pressure,  of  maximum  value  V  '2  £, 
which  is  applied  to  the  circuit,  and  has  a  frequency  /.  It 
is  expressed  in  volts. 

R  is  the  resistance  of  the  circuit,  and  is  expressed  in 
ohms.  It  is  numerically  equal  to  the  product  of  the  im- 
pedance by  the  cosine  of  <£. 

L  is  the  inductance  of  the  circuit,  and  is  expressed  in 
henrys. 

C  is  the  localized  capacity  of  the  circuit,  and  is  expressed 
in  farads. 

2  njL  is  the  inductive  reactance  of  the  circuit,  and  is 
expressed  in  ohms. 

i 

27T/C 

is  the  capacity  reactance,  or  capacitance,  of  the  circuit,  and 
is  expressed  in  ohms. 


is  the  reactance  of  the  circuit,  and  is  expressed  in  ohms  and 
usually  represented  by  X.  It  is  numerically  equal  to  the 
product  of  the  impedance  by  the  sine  of  </>. 


is  the  impedance  or  apparent  resistance  of  a  circuit,  and  is 
expressed  in  ohms  and  usually  represented  by  Z. 


or   |, 


the  reciprocal  of  the  impedance,  is  the  admittance  of  the 
circuit,  and  is  represented  by  F.     It  is  expressed  in  terms 


ALTERNATING-CURRENT   CIRCUITS. 


73 


of  a  unit  that  has  never  been  officially  named,  but  which  has 
sometimes  been  called  the  mho.  There  are  two  compo- 
nents of  the  admittance,  as  shown  in  Fig.  46. 

The  conductance  of  a  circuit,  usually  represented  by  g, 
is  that  quantity  by  which  E  must  be  multiplied  to  give  the 


Fig.  46. 


component  of  I  parallel  to  E.     It  is  expressed  in  the  same 
units  as  the  admittance,  and  is  numerically  equal  to 


cos  j> 
Z 


or    Y  cos 


but 


,       R    .  R 

cos  0  =  -  ,  hence  g  =    — 


The  susceptance  of  a  circuit,  represented  by  6,  is  that 
quantity  by  which  E  must  be  multiplied  to  give  the  com- 
ponent of  /  perpendicular  to  E.  It  is  measured  in  the  same 
units  as  the  admittance,  and  is  numerically  equal  to 


or    Y  sin  <£, 


but 


sin  <£.=  —  ,  hence    b  =  —  . 


Admittance  may  then  be  expressed  as 


Y  =  Vg2  +  b\ 

It  should  be  noticed  that  while  admittance  is  the  recip- 
rocal of  impedance,  conductance  is  not  the  reciprocal  of 


74 


ALTERNATING-CURRENT    MACHINES. 


resistance,  nor  is  susceptance  the  reciprocal  of  reactance. 
This  becomes  evident,  upon  considering  numerical  values 
in  connection  with  the  impedance  right-angled  triangle, 
e.g.  3,  4  and  5  for  the  sides. 

33.  Representation  of  Impedance  and  Admittance  by 
Complex  Numbers.  —  The  problem  of  determining  current, 
voltage  and  phase  relations  in  alternating-current  circuits 
may  be  solved  graphically,  by  means  of  vector  diagrams, 
or  trigonometrically.  To  facilitate  the  solution  of  particular 
problems  by  the  latter  method,  use  is  made  of  complex 
numbers. 

In  Fig.  47,  let  I  be  the   current  produced  in  a  circuit 

by  the  harmonic 
E.M.F.,  E,  the  cur- 
rent lagging  behind 
the  electromotive 
force  by  the  angle  <£. 
Taking  the  rect- 
angular reference 
.  axes  x  and  y  as 
shown,  both  E  and 
/  may  be  resolved 
into  components  along  them.  Let  the  symbol  j  be  placed 
before  the  ^-components,  thus  distinguishing  them  from  the 
^-components.  Then 


Fig.  47. 


and 


/  = 


je2 
ji2, 


the  plus  sign  indicating  vector  addition  at  right  angles  of  the 
x  and  y  components  respectively.  But  E  may  also  be 
resolved  into  a  component  in  phase  with  the  current  and 


ALTERNATING-CURRENT    CIRCUITS.  75 

into  another  at  right  angles   thereto,  that  is,  it  may  be 
expressed  as 

E  =  RI  +  JXI, 

and  substituting  the  values  of  E  and  7,  there  results 
el  +  je2  =  Rit  +  jRi2  +  jXit  +  ?Xi2. 

Both  RI  and   XI  may  be  resolved  into  components  along 
the  axes  of  reference  as  indicated,  and  hence  it  follows  that 

€l  =  R^  -  Xi2 

and  e2  =  Ri2  +  Xit. 

Then    R^  -  Xi2  +  jRi2  +  jXi±  =  Ri,  +jRi2  +  jX^  +  fXit. 
Therefore  -  Xi2  =  fXi, 

or  f  =  —  i 

and  ;  =  V  —  i, 

which  is  therefore  the  interpretation  of  the  symbol  /,  as 
already  denned. 

From  the  foregoing, 

E 


I  = 


R  +JX' 


but  7=|,  (§32) 

hence  the  impedance  Z  may  be  properly  represented  by 
R  +  jX. 

Admittance,  being  the  reciprocal  of  impedance,  may  then 

be  represented  by  -    — — ,  and  multiplying  both  numerator 
R  +  jX 

and  denominator  by  R  —  jX,  there  results, 

R-jX  R  -  jX    _  R  -  JX 

(R  +  JX)  (R  -  JX)       R2  -  ?X*      R2  +  X2  ' 


76  ALTERNATING-CURRENT    MACHINES. 


Separating,    Y  =  -  j 


7?  Y^ 

But  ^        "^  =  £    and     ~Z~2  =  &*  (§32) 

Hence  the  admittance  F  is  to  be  represented  by  g  —  ;7>. 

34.  Instantaneous  Current  in  a  Circuit  Having  Induct- 
ance, Capacity  and  Resistance.  —  In  §  20  an  expression  was 
derived  for  the  instantaneous  current  produced  -by  a  har- 
monic E.M.F.  in  a  circuit  having  inductance  and  resistance, 
and  in  §  30  a  similar  expression  was  derived  for  a  circuit 
having  capacity  and  resistance.  Proceeding  along  the  same 
lines,  a  general  expression  could  be  obtained  for  the  instan- 
taneous current  produced  by  a  harmonic  E.M.F.  in  any 
alternating-current  circuit,  that  is;  in  one  having  inductance, 
capacity  and  resistance.  This  method,  however,  is  rather 
cumbersome,  and  a  simpler  one  is  given  as  follows: 

The  harmonic  E.M.F.  is  represented  by  Emejut,  an 
expression  which  results  from  the  use  of  Maclaurin's  Series, 
that  is, 


+  jj[/"(*)],-o  +  £[/"'(*)!.-„  +  ... 

where  f'(x),  /"(#),  /"'(#)>  •  •  •  •  are    the  respective  deriva- 
tives of  f(x). 

When  the  function  is  sin  6, 

6s       65       (j1 
sin  0  =  6  -  .-  +      -_  +  ... 

|3       15       |Z 
and  when  the  function  is  cos  0, 

cos  6  =  i  -  ,-  +  , ,-  +  ... 


ALTERNATING-CURRENT    CIRCUITS.  77 

When,  however,  the  function  is  e?e,  then 


_ 

—     ~  i     '    i       '    i       *    i      ^  i       *    i^          i       i    •  •  • 

l£     fe       fe       It       H       &       fe. 

Remembering  that  f  =  —  i,  f  =  i,  f  =  —  i,  .  .  .  this 
becomes 

02      0*      6'P  .r,       6s   ,   6>5      ^7 

^=  i  -  ,-  +  i  —  \2  +  -  •  •  +  7  0  -  r  +  r  ~  r 
12  .14      1$  |3      IS      1Z 

Hence  ^'*  «  co^  0  +  j  sin  0. 

Multiply  through  by  Em  and  replace  0  by  &>/,  then 
Ems?™*  =  Em(cosa)t  +  ysinw/), 

which  is  evidently  a  proper  expression  for  a  harmonic  E.M.F. 
Consider  a  circuit  having  a  resistance  R,  a  capacity  C 
and  an  inductance  L.  The  E.M.F.  impressed  upon  this 
circuit  must  be  of  such  magnitude  as  to  neutralize  both  the 
counter  E.M.F.  of  self-induction  and  the  E.M.F.  of  di- 
electric polarization,  and  also  send  the  instantaneous 
current  /'  through  the  resistance.  Therefore 

E'  =  £/  +  £/  +  Er' 
or  Em  *"'  =L^+ 

Since  the  current  is  of  the  same  character  as  the  impressed 
E.M.F.,  it  may  be  represented  by  Be?wt,  where  B  is  a 
constant  to  be  determined.  Then 


and 

and     fl'dt  =  B  A 
«/                 «/ 

f  -#•*?* 

78  ALTERNATING-CURRENT    MACHINES. 

Substituting  these  values  in  (i),  there  results 
-    -—*** 


Hence     B= En 


But  /'  =  Be''"'  =*  ^  (cos  w/  +  /  sin 

and  substituting  value  of  B,  there  results 


/'  = 


[cos  cut  +  /  sin  tot], 

Em 


—    (R  cos  (ut  +\coL M  sin  w*J 

~  wC/ 
yYl?  sin  o>/  —    wL  —  —     cos  tot  \    . 


ALTERNATING-CURRENT    CIRCUITS.  79 

Assuming  the  impressed  harmonic  E.M.F.  as  a  simple  sine 
function,  then  only  the  second  part  within  the  bracket  of 

this  expression  need  be  taken;  hence 

/  _  \  -i 

•(2) 


_  \R  sin  wt  -  Li  -  ^COS  tjt\. 
T  Y  L  \  uLI  J 


m 

R 

Now  let  an  angle  (j>  be  chosen  so  that 

—  — 
cut 


, 
tan  <    = 


-w~ 


- 


then         coL  -  -      =        R*  + 


and  12  =  \/  R2 

Substituting  these  in  (2), 
2  + 


\ 


[sin  ft>/  cos  ^>  —  cos  wt  sin 


or  /'  -  -7=J===  sin  (at  -  ft,  (3) 


which  is  the  required  expression  for  the  instantaneous 
current  produced  by  a  harmonic  E.M.F.  in  a  circuit  con- 
taining inductance,  capacity  and  resistance. 

When  L  «  o,  the  expression  reduces  to  the  form  given 
in  §  30;  and  when  the  circuit  has  no  capacity  with  respect 

to  itself,  the  term  —  drops  out,  and  the  expression  reduces 
coC 

to  the  form  given  in  §  20.  It  follows,  then,  that  equation 
(3)  may  be  applied  to  any  alternating-current  circuit. 


8O  ALTERNATING-CURRENT   MACHINES. 

35.  Resonance.  —  An  electrical  circuit  is  said  to  be 
resonant,  or  in  resonance  with  an  impressed  E.M.F.,  when 
the  natural  period  of  that  circuit  and  the  period  of  the 
E.M.F.  are  the  same.  The  natural  period  of  the  circuit 
is  the  reciprocal  of  that  frequency  at  which  the  current  is  a 
maximum.  By  reference  to  the  formula 

E 


I  = 


\/  R*  + 


2  7T/C 


it  becomes  evident  that  the  maximum  current  is   —  ,  which 

R 

occurs  when 

2  7T/L  —     --  —  ;     =   O, 
2  71  JC 

that  is,  when  the  capacity  and  the  inductance  are  so  pro- 
portioned that  their  reactances  are  equal.  From  this 
relation,  it  follows  that  the  critical  frequency  at  which 
resonance  occurs  is 


Lc 

and  that  the  natural  period  of  the  circuit  is  2  n  \/LC. 

To  show  the  current  values  for  different  frequencies,  a 
curve  as  in  Fig.  48  may  be  drawn.  It  is  plotted  for  a  series 
circuit,  having  5  ohms  resistance  and  an  inductance  of 
0.422  henry  and  a  capacity  of  6  microfarads;  upon  which 
circuit  is  impressed  a  harmonic  E.M.F.  of  100  volts.  The 
frequency  of  the  impressed  E.M.F.  required  for  resonance 
is  seen  to  be  100  ~.  At  this  frequency  the  potential  dif- 

ference between  the  terminals  of  the  condenser  =  -  - 

27T/C 

=  5300  volts,  and  that  across  the  inductance  coil  is  2  KJLI 
=*  5300  volts  also,  whereas  only  100  volts  is  impressed  upon 


ALTERNATING-CURRENT    CIRCUITS. 


81 


the  circuit.  Hence  when  resonance  occurs  m  a  circuit  in 
which  the  capacity  and  the  inductance  are  in  series,  the 
potential  difference 
across  either  may 
rise  to  such  a  value 
as  to  puncture  the 
insulation  of  the 
apparatus.  If  the 
capacity  and  induc- 
tance be  in  parallel, 
enormous  currents 
may  flow  between 
the  two.  This  is 


FREQUENCY 

Fig.  48. 


because  the  two  are  balanced,  and  the  one  is  at  any  time 
ready  to  receive  the  energy  given  up  by  the  other;  and  a 
surging  once  started  between  them  receives  periodical  in- 
crements of  energy  from  the  line.  This  is  analogous  to 
the  well-known  mechanical  phenomena  that  a  number  of 
gentle,  but  well-timed,  mechanical  impulses  can  set  a  very 
heavy  suspended  body  into  violent  motion.  The  frequency 
of  these  impulses  must  correspond  exactly  to  the  natural 
period  of  oscillation  of  the  body.  In  this  parallel  arrange- 
ment, serious  damage  is  likely  to  result  from  resonance  in 
overloading  and  burning  out  the  conductors  between  the 
inductance  and  capacity. 

The  protection  of  transformers  and  other  station  appara- 
tus against  high-potential  surges  coming  from  transmission 
lines  is  effected  by  the  use  of  choke  coils  interposed  between 
the  lines  and  the  station  wiring.  It  is  essential  for  proper 
protection,  that  the  electrostatic  capacity  of  this  wiring  be 
as  small  as  possible  and  that  the  choke  coil  have  as  low  an 
inductance  as  will  allow  the  lightning  arrester  to  take  up 


82  ALTERNATING-CURRENT    MACHINES. 

the  discharge,  so  that  the  frequency  of  resonance  will  be 
raised,  thus  decreasing  the  liability  of  picking  up  destructive 
voltages  from  line  impulses  or  lightning  discharges. 

36.  Damped  Oscillations.  —  When  a  condenser  is  dis- 
charged through  a  circuit  having  resistance  and  inductance, 
an  oscillatory  current  flows,  the  maximum  values  of  which 
decrease  logarithmically.  The  ratio  of  two  successive 

a 

maximum  values  can  be  shown  equal  to  e  2/,  where  a  is  the 

r> 

damping  factor  and  is  equal  to  —  -  ,  R  being  the  high-fre- 

2  -L/ 

quency  resistance  of  the  circuit.     The  entire  exponent,  —  - 
is  called  the  logarithmic  decrement,  and  is  represented  by  d. 

Hence  d  =  -^l, 


and  replacing  /  by  -  y  ~—  ,  (§  35  ) 

2  7T          -LC 


there  results  d  =   —     v   _ 

2     *  L 

The  effective  current  value  of  a  train  of  damped  oscilla- 


/\ 


Fig-   49- 


tions,  one  of  which  is  shown  in  Fig.  49,  can  be  deduced  by 
considering  the  energy,  stored  in  the  condenser  at  each 


ALTERNATING-CURRENT    CIRCUITS.  83 

charge  and  discharged  n  times  per  second,  to  be  consumed 
in  heating  the  conductors.     Then  from  §  26, 


and  substituting  the  value  of  R  in  the  expression  for  d,  there 
is  obtained 


./C  . 
VL' 


and  since  Q  =  EmC, 


r-.  /x-1 

—         nnj±m  LX    t  /C- 

~T^VZ' 

Hence  the  effective  value  of  the  current  is 
/  =E™\ 


The  natural  frequency  of  a  circuit  in  which  a  decaying 
oscillatory  current  flows  is  dependent  only  upon  R,  C  and 
L,  and  may  be  obtained  from  the  formula 


where     -  >  a\ 


When  a2  >  -—  -  ,  the  current  is  unidirectional  and  decreases 

.L/O 

as  in  Fig.  38. 

37.  Polygon  of  Impedances.  —  Consider  a  circuit  having 
a  number  of  pieces  of  apparatus  in  series,  each  of  which 
may  or  may  not  possess  resistance,  inductance,  and  capacity. 
There  can  be  but  one  current  in  that  circuit  when  a  pressure 


84  ALTERNATING-CURRENT   MACHINES. 

is  applied,  and  that  current  must  have  the  same  phase 
throughout  the  circuit.  The  pressure  at  the  terminals  of 
the  various  pieces  of  apparatus,  necessary  to  maintain 
through  them  this  current,  may,  of  course,  be  of  different 
magnitude  and  in  the  same  or  different  phases,  being 
dependent  upon  the  values  of  R,  L,  and  C.  Therefore  to 
determine  the  pressure  necessary  to .  send  a  certain  alter- 

R,    L,    C, 


WWVW '    ' 


Fig.  50. 


nating  current  through  such  a  series  circuit,  it  is  but  neces- 
sary to  add  vectorially  the  pressures  needed  to  send  such  a 
current  through  the  separate  parts  of  the  circuit.  This  is 
readily  done  graphically  although  in  many  cases  the  various 
quantities  may  be  of  such  widely  different  magnitudes  that 
it  will  be  found  more  convenient  to  make  use  of  trigono- 
metrical expressions  and  methods. 

Fig.   50  shows  the  pressures  (according  to  §  31)  neces- 
sary to  send  the  current  /  through  several  pieces  of  ap- 


ALTERNATING-CURRENT   CIRCUITS.  85 

paratus,  and  the  combination  of  these  pressures  into  a 
polygon  giving  the  resultant  pressure  E  necessary  to  send 
the  current  /  through  the  several  pieces  in  series.  In 
these  diagrams,  impedance  is  represented  by  the  letter  Z. 
C^  and  C3  are  localized,  not  distributed  capacities. 

For  practical  purposes,  the  quantity  7,  which  is  common 
to  each  side  of  the  triangle,  may  be  omitted  ;  and  merely 
the  impedances  may  be  added  vectorially  in  a  "polygon  of 
impedances,"  giving  an  equivalent  impedance,  which,  when 
multiplied  by  /,  gives  E. 

Inspection  of  the  figure  shows  that  the  analytical  ex- 
pression for  the  required  E  is 


The  pressure  at   the  terminals  of  any  single  part  of  the 
circuit  is 


It  is  evident  that 

£i  +  E2  + >  E, 

and  it  is  found  by  experiment  that  the  sum  of  the  potential 
differences,  as  measured  by  a  voltmeter,  in  the  various 
parts  of  the  circuit,  is  greater  than  the  impressed  pressure. 

38.  A  Numerical  Example  Applying  to  the  Arrangement 
Shown  in  Fig.  50.  —  Suppose  the  pieces  of  apparatus  to 
have  the  following  constants  : 


86 


ALTERNATING-CURRENT    MACHINES. 


^  =    85  ohms,      Zj  =  .25  henry,      C±  =  .000018  farad  (18  mf.) 

RI  =    40  ohms,      Z2  =  .3    henry,       

Cs  =  .000025  farad, 

^4  =  100  ohms.       

With  a  frequency  of  60  cycles  —  whence  0  =  377  —  it  is 
required  to  find  the  pressure  necessary  to  be  applied  to  the 
circuit  to  send  10  amperes  through  it. 


The  completion  of  the  successive  parallelograms  in 
Fig.  51,  is  equivalent  to  completing  the  impedance  poly- 
gon, and  the  parts  are  so  marked  as  to  require  no  explana- 
tion. The  solution  shows  that  the  equivalent  impedance, 
^=229.5  ohms,  that  the  equivalent  resistance  (=  actual 
resistance  in  series),  R  =  22$  ohms,  that  the  equivalent  re- 
actance is  condensive  and  equals  46.2  ohms,  and  that  <£  = 


ALTERNATING-CURRENT   CIRCUITS.  87 

11.55°  of  lead.     Hence  the  pressure  required  to  send   10 
amperes  through  the  circuit  is 

E  =  10  x  229.5  =  2295  volts. 
To  obtain  the  same  results  analytically 

E  =  10  V[85  +  40+  ioo]2+  [(94.2  + 113.1)  -  (147.3  +  io6.2)]2 
E  =  2295.  volts. 

The  voltages  at  the  terminals  of  the  various  pieces  of  ap- 
paratus are  : 


.2  —  147.  3)2    =1001  volts, 


z=  10  V4Q2+  1  13.  i2  =1200 

s=  10  Vo2+  io6.22  =1062 

£"4  =  10  Vioo2  +'  o2  =  1000 


which  is  greater  than  .£"  =  2195  volts,  showing  that  the 
numerical  sum  of  the  pressures  is  greater  than  the  im- 
pressed pressure  ;  while  the  vectorial  sum  of  the  separate 
pressures  is  equal  to  the  impressed  pressure. 

39.  Polygon  of  Admittances.  —  If  a  group  of  several 
impedances,  Z^  Zii?  etc.,  be  connected  in  parallel  to  a 
common  source  of  harmonic  E.M.F.  of  E  volts,  their 
equivalent  impedance  is  most  easily  determined  by  con- 
sidering their  admittances  F1?  F2,  etc.  The  currents  in 
these  circuits  would  be 


The  total  current,  supplied  by  the  source,  would  be  the 
vector  sum  of  these  currents,  due  consideration  being  given 
to  their  phase  relations.  Calling  this  current  /,  the  equation 
I—EY  can  be  written,  where  Fis  the  equivalent  admit- 


88 


ALTERNATING-CURRENT   MACHINES. 


tance  of  the  group.  To  determine  F,  a  geometrical  addition 
of  F15  F2,  etc.,  must  be  made,  the  angular  relations  being 
the  same  as  the  phase  relations  of  Iv  72,  etc.,  respectively. 
The  value  of  the  equivalent  admittance  may  therefore  be 
represented  by  the  closing  side  of  a  polygon,  whose  other 
sides  are  represented  in  magnitude  by  the  several  admit- 
tances Yv  F2,  etc.,  and  whose  directions  are  determined 
by  the  phase  angles  of  the  currents  Iv  /2,  etc.,  flowing 
through  the  admittances  respectively. 

Fig.  52  is  a  polygon  of  admittances  showing  the  method 
of  obtaining  the  equivalent  admittance  graphically  for  a 


Fig.  52- 


number  of  admittances  in  parallel.     The  equivalent  admit- 
tance may  also  be  determined  analytically,  since 


+ 


and  hence  it  follows  that  the  current 


/  =  E  V(gl  +  g,  +  .  .  .  )2  +  (b,  +  b2  +  .  .  .  )2, 

where  glt  g2,  .  .  .  and  bv  b2,  .  .  .  are  the  respective  con- 
ductances and  susceptances  of  the  various  pieces  of  appa- 
ratus. 

The  instantaneous  value  of  the  current  in  the  main  cir- 
cuit is  equal  to  the  sum  of  the  instantaneous  current  values 


ALTERNATING-CURRENT   CIRCUITS. 


89 


R,  1,0, 


in  the  branch  circuits,  but  since  their  maximum  values  occur 
at  different  times,  the  sum  of  the  effective  values  of  current 
in  the  branches  generally  exceeds  the  effective  current  value 
in  the  mains. 

As  a  numerical  example  on  the  foregoing,  consider  the 
same  apparatus  as  was  used  in  the 
preceding  example  (§  38),  to  be  arranged 
in  parallel,  as  in  Fig.  53.  It  is  required 
to  find  the  current  that  will  flow  through 
the  mains  when  a  60  ~  alternating  E.M.F. 
of  ten  volts  is  impressed  upon  the 
circuit. 

7?  Y 

Remembering  that  g  =  —  and  that  b  =  —  and  referring 

Z*  Zj 

to  §  38  for  the  numerical  values,  the  conductances  and 
susceptances  of  the  branch  circuits  are 


Si- 

82- 


loo.r 
40 

I202 


.00848 
.00278 


120 
106.2 

I06.22 


=  -  -0053 
=        .00786 
—  .00942 


100* 

Adding  algebraically, 
g    =  .02126 


b    =  -  .00686 


Then      Y  =  V/(.02I26)2  +  (—  .oo686)2  =  .0224. 
Hence    /   =  EY  =  10  X  .0224  =  .224  amperes. 


ALTERNATING-CURRENT    MACHINES. 


The  phase  of  /  is  given  by 


tan~ 


_,  b  -  .00686  _ 

—  —  tan     —————  —  — 


.02126 


S3'. 


the  negative  sign  indicating  that  the  current  leads  the 
E.M.F.  The  admittance  of  each  branch  circuit  and  the 
value  and  phase  of  the  current  therein,  may  be  calculated 
by  proceeding  in  a  similar  manner. 

40.  Impedances  in  Series  and  in  Parallel.  —  If  a  circuit 
have  some  impedances  in  series  and  some  in  parallel,  or  in 
any  series  parallel  combination,  the  equivalent  impedance 
can  always  be  found  by  determining  the  equivalent  impe- 
dances of  the  several  groups,  and  then  combining  these 
resulting  impedances  to  get  the  total  equivalent  impedance 
sought.  To  illustrate,  a  problem  will  be  worked  out  in 
detail. 

Let  it  be  required  to  determine  the  values  and  phases  of 
the  currents  in  the  main  and  in  each  of  the  four  branch 


Fig.  54. 

circuits,   A,  B,   C  and   D  of  the   combination  shown  in 
Fig.  54,  when  the  main  terminals  are  connected  to  a  2oo-volt 
25-cycle  alternator. 
The  constants  of  the  apparatus  and  the  results  of  the 


ALTERNATING-CURRENT   CIRCUITS  91 

various  steps  in  the  calculation  are  given  below  in  tabulated 
form,  and  require  no  explanation. 


A 

B 

C 

D 

R 

5° 

20 

0 

150 

ohms 

L 

i 

-5 

0 

6 

henrys 

C 

.00002 

0 

.00001 

.000008 

farads 

<*L 

157.08 

78.54 

o 

942.48 

i 

318.31 

O 

636.62 

795.78 

X 

-    161.23 

78.54 

-  636.62 

146.70 

z 

168.80 

81.05 

636.62 

209.80 

0 

-    72°  46' 

75°  43' 

-90° 

44°  38' 

F 

•00593 

.01234 

.001571 

.00477 

g 

.001758 

.00305 

o 

.00339 

b 

—   .00566 

.01195 

-  .001571 

•0°335 

SA+B 

=  .00481 

gc+D      =-°°339 

bA  +  B 

=  .00629 

bc+D      =  .00 

178 

YAB 

=  .00792 

YCD        =  .00383 

$AB 

-  52°  36' 

<j>CD        =  27°  42' 

ZAB 

=  126.3 

ZCD        =  261 

.2 

RAB 

=    76.74 

RCD        =  23i 

•3 

XAB 

=  100.3 

XCD           =  I21 

.4 

Rt    =  308.0  = 

total  equiv 

.  resistance 

,. 

Xt  =  221.7   = 

total  equiv 

.  reactance 

Zt    =  379-5   - 

total  equiv 

.  impedance 

I]    =.-529     = 

current  in 

mains 

<t>t    =  35°  45'  = 

=  phase  of 

If 

F       — 
*^AK 

ZAB  lt  =  66'81 

F          —  7      T    - 

^CD     ~  ^CDlt  ~ 

138.17 

*  A    =  -396 

I 

c  =  >2I7 

••  .824 


ID  =  .649 


It  is  evident  that  the  sum  of  the  potential  differences 
across  the  two  groups  is  greater  than  the  impressed  E.M.F., 


92  ALTERNATING-CURRENT   MACHINES. 

and  that  the  arithmetical  sum  of  the  currents  in  the  branch 
circuits  exceeds  the  total  current  flowing  in  the  main  cir- 
cuit. The  relative  magnitudes  and  the  phases  of  the 


Fig.  55- 

various  currents  and  E.M.F.'s  in  the  different  circuits  are 
represented  in  Fig.  55,  which  also  serves  as  a  rough  check 
upon  the  calculations. 

PROBLEMS. 

1.  A  60  ~w  alternating  E.M.F.  of  200  volts  maximum  is  impressed 
upon  a  circuit  having  120  ohms  resistance,  an  inductance  of  i  henry  and 
a  capacity  of  25  microfarads.     Determine  the  value  and  phase  of  the 
current  flowing  in  the  circuit. 

2.  What  are  the  values  of  X,  Z,  Y,  b,  and  g  in  the  preceding  problem  ? 


PROBLEMS.  93 

3.  If  a  harmonic  E.M.F.  of  220  volts  (effective)  is  impressed  upon  a 
circuit,  producing  a  current  of  20  amperes  lagging  30°,  what  will  be  the 
resistance  and  reactance  of  the  circuit? 

4.  Find  the  instantaneous  current  value  in  the  circuit  of  Problem  i, 
if  seconds  after  impressing  the  harmonic  E.M.F.  upon  it. 

5.  What  is  the  resonant  frequency  of  a  circuit  having  10  microfarads 
capacity  and  an  inductance  of  .352  henry  ?    What  will  be  the  drop  across 
the  condenser  at  resonance,  when  10  amperes  flow  through  it  ? 

6.  A  condenser  of  .003  mf.  capacity  is  charged  20  times  per  second 
to  a  potential  of  1000  volts.     What  is  the  mean  effective  current  value 
of  the  discharge  in  a  circuit  having  an  inductance  of  2  millihenrys,  if 
the  decrement  of  the  oscillations  is  .2? 

7.  Determine  the  pressure  required  to  send  10  amperes  through  the 
circuit  shown  in  Fig.  50,  if  the  frequency  is  25  ~~  per  second,  the  values 
of  the  resistances,  inductances  and  capacities  being  the  same  as  in  §  38. 
Give  also  a  graphic  solution. 

8.  In  the  arrangement  shown  in  Fig.  53,  using  the  same  impedances 
as  in  the  preceding  problem,  what  current  will  traverse  the  mains,  if  a 
25-cycle  alternating  E.M.F.  of  10  volts  is  impressed  thereon  ? 

9.  If  the  200- volt  25  ~w  alternator  of  §  40  were  replaced  by  another  of 
the  same  voltage  but  of  6o^~  frequency,  what  would  be  the  values  and 
phases  of  the  currents  in  the  main  and  in  the  branch  circuits?" 

10.  Let  the  impedance  D  of  the  preceding  problem  be  disconnected 
from  the  circuit.     Then  determine  the  values  and  phases  of  the  currents 
in  the  main  and  in  the  branch  circuits  A  and  B.     Construct  a  vector 
diagram  showing  the  voltage  and  current  relations. 


94  ALTERNATING-CURRENT   MACHINES. 


CHAPTER   V. 

ALTERNATORS. 

41.  Alternators.  —  An  alternator  is  a  machine  used  for 
the  conversion  of  mechanical  energy  into  electrical  energy, 
which  is  delivered  as  alternating  current,  either  single- 
phase  or  polyphase.  Alternators,  like  direct-current  gen- 
erators, have  field-magnets  and  an  armature;  but  the 
commutator  of  the  direct-current  machine  is  replaced  in 
alternators  by  slip-rings,  which  deliver  alternating  current 
to  brushes  rubbing  upon  them  when  the  armature  rotates, 
or  receive  direct  currents  for  exciting  the  field-magnets 
when  they  rotate.  A  polyphase  alternator  produces  two 
or  more  single-phase  alternating  E.M.F.'s,  which  in  opera- 
tion send  currents  in  circuits  which  may  or  may  not  be  in 
electrical  connection  with  each  other.  The  only  relation 
between  these  E.M.F.'s  is  that  of  time,  that  is,  they  differ 
in  phase.  These  phase  differences  depend  upon  the 
relative  positions  of  the  armature  windings  and  may  be 
anything  from  o°  to  360°,  but  it  is  customary  to  place  them 
so  as  to  produce  E.M.F.'s  differing  symmetrically  in  phase. 
In  two-phase  or  four-phase  alternators,  the  E.M.F.'s  are 
90°  apart,  in  three-phase  alternators  120°  apart,  and  in  six- 
phase  alternators  60°  apart. 

As  it  is  the  relative  motion  of  the  armature  and  field- 
magnets  which  is  essential  in  the  generation  of  E.M.F.,  it 
is  quite  as  common  to  have  the  field-magnets  of  an  alternator 


ALTERNATORS.  95 

revolve  inside  the  armature  as  to  have  the  armature  revolve; 
in  fact,  nearly  all  large  alternators  are  of  the  revolving- 
field  type,  the  revolving-armature  type  being  now  generally 
restricted  to  smaller  units.  The  chief  advantage  of  the 
revolving-field  type  alternator  is  that  it  avoids  the  collection 
of  high-tension  currents  through  brushes,  since  the  arma- 
ture connections  are  fixed,  and  only  low-tension  direct 
current  need  be  fed  through  the  rings  to  the  field-coils. 
Other  advantages  are  increased  room  for  armature  insula- 
tion, and,  in  polyphasers,  the  avoidance  of  more  than  two 
slip-rings.  Revolving-field  alternators  have  been  con- 
structed to  generate  25,000  volts,  whereas  the  E.M.F. 
produced  in  the  revolving-armature  type  is  practically 
limited  to  5000  volts.  In  a  few  instances,  notably  at 
Niagara  Falls,  the  field-magnets  revolve  outside  the  arma- 
ture. 

Besides  the  two  types  mentioned,  there  is  the  inductor 
type  of  alternator,  which  has  both  its  armature  and  its 
field  windings  stationary  and  has  an  iron  rotating  member 
termed  an  inductor.  In  this  type  there  are  neither  brushes, 
collector  rings,  nor  moving  electrical  circuits. 

It  is  necessary  that  all  but  the  very  smallest  alternators 
should  be  multipolar  to  fit  them  to  commercial  require- 
ments. For  alternators  must  have  in  general  a  frequency 
between  25  and  125  cycles  per  second;  the  armature 
must  be  large  enough  to  dissipate  the  heat  generated  at 
full  load  without  its  temperature  rising  high  enough  to 
injure  the  insulation;  and  finally,  the  peripheral  speed  of 
the  armature  cannot  safely  be  made  to  exceed  greatly  a 
mile  a  minute.  With  these  restrictions  in  mind,  and 
knowing  that  a  point  on  the  armature  must  pass  under 
two  poles  for  each  cycle,  it  becomes  evident  that  alterna- 


96  ALTERNATING-CURRENT   MACHINES. 

tors  of  anything  but  the  smallest  capacity  must  be  multi- 
polar. 

42.   Electromotive  Force  Generated.  —  In  §  13,  vol.  i.,  it 

was  shown  that  the  pressure  generated  in  an  armature  is 

;     -  £C 

'         -  --* 


where  p  =  number  of  pairs  of  poles, 

<£  =  maxwells  of  flux  per  pole, 
V  =  revolutions  per  minute, 

and  S  =  number  of  inductors. 

In  an  alternating  circuit  E  =  k^E^,  where  \  is  the 
form-factor,  i.e.,  the  ratio  of  the  effective  to  the  average 
E.M.F.  Hence  in  an  alternator  yielding  a  sine  wave 
E.M.F., 

E  =  2.22  p&S—  io"8. 
60 

Inasmuch  as  p  —  -  represents  the  frequency,  /, 
60 

E   =    2.22  <*>£/  I0~8. 

An  alternator  armature  winding  may  be  either  concen- 
trated or  distributed.  If,  considering  but  a  single  phase, 
there  is  but  one  slot  per  pole,  and  all  the  inductors  that  are 
intended  to  be  under  one  pole  are  laid  in  one  slot,  then 
the  winding  is  said  to  be  concentrated,  and  if  the  inductors 
are  all  in  series  the  above  formula  for  E  is  applicable. 
Nearly  all  engine-driven  alternators  have  six  slots  per  pole 
although  twelve  slots  per  pole  are  used  when  the  output 
per  pole  is  large  and  a  long  armature  is  undesirable.  If 


ALTERNATORS.  97 

now  the  inductors  be  not  all  laid  in  one  slot,  but  be  dis- 
tributed in  n  more  or  less  closely  adjacent  slots,  the  E.M.F. 

generated  in  the  inductors  of  any  one  slot  will  be  -  of  that 

generated  in  the  first  case,  and  the  pressures  in  the  differ- 
ent slots  will  differ  slightly  in  phase  from  each  other,  since 
they  come  under  the  center  of  a  given  pole  at  different 
times.  The  phase  difference  between  the  E.M.F.  gener- 
ated in  two  conductors  which  are  placed  in  two  successive 
armature  slots,  depends  upon  the  ratio  of  the  peripheral 
distance  between  the  centers  of  the  slots  to  the  peripheral 
distance  between  two  successive  north  poles  considered  as 
360°.  This  phase  difference  angle 

width  slot  -+-  width  tooth 

9  =   -: 7OO. 

circumference  armature 
no.  pairs  poles 

If  the  inductors  of  four  adjacent  slots  be  in  series,  and 
if  the  angle  of  phase  difference  between  the  pressures 
generated  in  the  successive  ones  be  <£,  then  letting  Elt  E^ 
E3,  and  E±  represent  the  respective  pressures,  which  are 


Fig.   56. 


supposed  to  be  harmonic,  the  total  pressure,  Et  generated 
in  them  is  equal  to  the  closing  side  of  the  polygon  as 
shown  in  Fig.  56.  Obviously  E  <  El  -f  E2  -f  Ez  -\-  Ef 
If  the  winding  had  been  concentrated,  with  all  the  induc- 
tors in  one  slot,  the  total  pressure  generated  would  have 
been  equal  to  the  algebraic  sum. 


ALTERNATING-CURRENT   MACHINES. 


The  ratio  of  the  vector  sum  to  the  algebraic  sum  of 
the  pressures  generated  per  pole  and  per  phase  is  called  the 
distribution  constant.  Not  only  may  the  number  of  slots 
under  the  pole  vary,  but  they  may  be  spaced  so  as  to 
occupy  the  whole  surface  of  the  armature  between  succes- 
sive pole  centers  (the  peripheral  distance  between  two 
poles  is  termed  the  pole  distance),  or  they  may  be  crowded 

together  so  as  to 
occupy  only  one- 
half,  one-fourth,  or 
any  other  fraction 
of  this  space.  Both 
the  number  of  slots 
and  the  fractional 
part  of  the  pole  dis- 
tance which  they 
occupy  affect  the 
value  of  the  distri- 
bution constant.  A 


Ocfcupi 


dby 


3  Slots. 

4  Slots. 
Many 
Slots. 


Fig.   57. 


set  of  curves,  Fig. 
57,  has  been  drawn, 
showing  the  values  of  this  constant  for  various  conditions. 
Curves  are  drawn  for  one  slot  (concentrated  winding),  2, 
3,  4  slots  in  a  group,  and  many  slots  (i.e.,  smooth  core 
with  wires  in  close  contact  on  the  surface).  The  ordinates 
are  the  distribution  constants,  and  the  abscissae  the  frac- 
tional part  of  the  pole  distance  occupied  by  the  slots. 

The  distribution   constant,  k^  must  be  introduced  into 
the  formula  for  the  E.M.F.  giving 

•£          -8 

'6oIC 
or,  for  sine  waves, 

E  =  2.22  k&Sf  io~8. 


ALTERNATORS. 


99 


4-  POLE.  \ 

f  SINGLE  PHASE.          \ 

CONCENTRATED. 

ADDITION  OF  DOTTED  WINDING  MAKES  IT 
TWO  PHASE 


43.  Armature  Windings.  —  There  are  separate  and 
distinct  windings  on  the  armature  core  of  a  polyphase 
alternator  for  each  phase,  and  these  may  each  be  separately 
connected  to  an  outside  circuit  through  a  pair  of  terminals, 
or  they  may  be  connected  to- 
gether in  the  armature  accord- 
ing to  some  scheme  whereby 
one  terminal  will  be  common 
to  two  phases.  Some  simple 
diagrams  of  the  armature 
windings  of  multipolar  alter- 
nators are  given  in  the  ac- 
companying figures.  Fig.  58 
shows  a  single-phase  concen- 
trated winding,  with  the  wind- 
ing necessary  to  render  it 
two-phase  indicated  by  dotted 
lines.  If  the  two  windings  be  electrically  connected  where 
they  cross  at  the  point  P,  the  machine  becomes  a  star- 
connected  four-phase  or  quarter-phase  alternator. 

Three-phase  alternators  might  be  provided  with  six  slip- 
rings  or  terminals,  thus  supplying  three  distinct  circuits 
with  single-phase  alternating  current,  or  with  four  slip-rings 
or  terminals,  one  of  which  should  be  connected  to  a  common 
return  wire  for  the  three  currents.  These  are  uncommon, 
however,  since  the  usual  practice  is  to  provide  only  three 
slip-rings  or  terminals,  each  connected  wire  acting  as  a 
return  path  for  the  currents  flowing  in  the  other  two.  It 
follows,  then,  that  the  current  in  one  wire  of  a  three-phase 
system  at  any  instant  is  equal  and  opposite  to  the  sum  of 
the  currents  in  the  other  two  wires  at  that  instant.  This  is 
shown  in  Fig.  59,  where  the  dotted  curve,  representing  the 


Fig.  58. 


IOO 


ALTERNATING-CURRENT    MACHINES. 


sum  of  the  two  current  curves,  is  exactly  equal  and  opposite 
to  the  third  current  curve. 

There  are  two  methods  of  connecting  the  armature  wind- 
ings of  three-phase  alternators  which  are  called  respectively 
F-and  A-connections.  In  the  first,  one  end  of  each  winding 


Fig.  59- 


is  connected  to  a  slip-ring  or  terminal;  the  other  ends  being 
joined  together  form  a  neutral  connection,  which  sometimes 


\ 


\ 


CONCENTRATED. 


CONCENTRATED. 


Fig.  60.  Fig.  61. 

is  connected  with  a  fourth  slip-ring  or  terminal  adapting  the 
alternator  for  use  with  a  three-phase,  four-wire  system.  In 
armatures  having  a  A-connection,  the  three  windings  are  con- 
nected together  in  series  to  forma  closed  circuit,  each  junction 
being  connected  to  a  slip-ring  or  terminal  post.  A  three-phase, 
F-connected,  concentrated  armature  winding  is  shown  in 
Fig.  60,  and  the  same  when  A-connected  is  shown  in  Fig.  61. 


ALTERNATORS.        ;       /;,  \ 

In  these  winding  diagrams  the  radial  lines  represent  the 
inductors,  and  the  other  lines  the  connecting  wires;  the 
inductors  of  different  phases  being  drawn  differently  for 
clearness.  Where  but  one  inductor  is  shown,  in  practice 
there  would  be  a  number  wound  into  a  coil  and  placed  in 
one  slot.  For  simplicity  all  the  inductors  of  one  phase  are 
shown  in  series.  Alternator  armatures  with  distributed 
windings  can  also  be  represented  diagrammatically  similar 
to  the  foregoing,  but  the  diagrams  become  very  complex 
when  there  are  many  slots  per  pole  per  phase.  For  sim- 


Fig.  62. 


plicity,  a  rectified  diagram  is  given,  Fig.  62  representing  the 
armature  winding  of  a  three-phase  alternator.  There  are 
five  slots  per  pole  per  phase. 

44.  Voltage  and  Current  Relations  in  Two-Phase  Systems. 
—  A  two-phase  alternator  may  be  considered  as  two  sepa- 
rate single-phase  alternators  of  the  same  size,  the  E.M.F.'s 
of  which  are  maintained  at  a  phase  difference  of  90°.  The 
maintenance  of  the  phase  relation  might  be  accomplished  by 
mounting  the  two  armatures  on  the  same  shaft  and  then 
placing  the  coils  in  the  same  relative  position  with  the  two 


MACHINES. 


field-magnets  displaced  from  each  other  by  ninety  degrees, 
or  with  the  field-magnets  in  the  same  relative  position  and 
the  armature  coils  displaced  by  ninety  degrees.  Let  these 
two  alternators  be  denoted  by  i  and  2,  Fig.  63,  and  assume 
that  the  E.M.F.  of  alternator  No.  2  lags  90°  behind  that 
generated  in  alternator  No.  i.  Then  the  time  variations 


Fig.  63. 

of  their  E.M.F.'s  may  be  graphically  represented  as  shown. 
This  condition  is  also  represented  by  the  vectors  E{  and 
E2,  lag  being  clockwise.  Let  the  effective  values  of  the 
E.M.F. 's  produced  in  either  armature  winding  be  E  volts 
and  the  effective  current  value  therein  be  /  amperes.  Then, 
since  the  two  circuits  of  a  two-phase,  four-wire  system  are 
electrically  distinct,  the  voltage  across  each  is  E  volts,  their 


Fig.  64- 


phase  relations  being  shown  by  the  vectors,  and  the  current 
in  each  line  wire  is  /  amperes. 


ALTERNATORS. 


103 


Now  consider  these  two  alternators  to  be  connected  as 
shown  in  Fig.  64,  thus  forming  a  two-phase,  three-wire 
system.  The  other  conditions  remaining  unaltered,  the 
E.M.F.'s  across  AB  and  EC  will  be  the  same  as  before  or 
E  volts,  and  the  current  flowing  in  A  or  C  will  similarly  be 
/  amperes.  The  voltage  across  AC  is  due  to  the  E.M.F.'s 
produced  in  both  alternators,  and  its  instantaneous  value 
is  equal  to  the  algebraic  sum  of  their  simultaneous  E.M.F.'s. 
The  curves  showing  the  time  variation  of  these  instan- 
taneous values  are  annexed,  and  EAC  is  seen  to  lag  45° 
behind  EAB.  These  conditions 
may  also  be  represented  by 
vectors  as  in  Fig.  65,  and  there- 
from 

EAC  =  EAB®  EBC  =  \/lE, 

which  is  therefore  the  voltage 
across  the  lines  A  and  C,  and 
it  lags  45°  behind  EAB. 

It  should  be  noted  that  if 
E2  leads  El  by  90°,  then  EAC 
will  lead  EAB  by  45°;  and  fur- 
ther, if  the  terminals  of  the  receiving  apparatus  be  re- 
versed, the  phases  of  the  E.M.F.'s  sending  current  through 
them  will  be  reversed.  Assuming  load  to  be  applied 
between  A  and  B,  and  B  and  C  only,  and  further  that 
the  circuits  are  balanced,  the  current  in  line  C  will  then 
lag  90°  behind  the  current  in  A,  as  shown  in  Fig.  65. 
Knowing  that  the  instantaneous  value  of  the  current  in 
line  B  is  equal  and  opposite  to  the  sum  of  the  instanta- 
neous current  values  in  lines  A  and  C,  its  value  and  phase 
may  be  determined  by  adding  —  IA  and  —  Ic  vectorially 


EAC 


104 


ALTERNATING-CURRENT    MACHINES 


as  shown.     Thus  In  is  seen  to  be  equal  to  \/2  /  and  to  lag 
225°  +  0  behind  EAB. 

45.  Voltage  and  Current  Relations  in  Three-Phase 
Systems.  —  Consider  a  three-phase,  F-connected  alternator 
to  consist  of  three  single-phase  genera- 
tors whose  E.M.F.'s  are  maintained  at  a 
successive  phase  displacement  of  120° 
(§  44),  their  external  connections  being 
as  shown  in  Fig.  66.  Let  the  directions 
of  the  E.M.F.'s  in  the  three  armature 
coils  as  their  axes  successively  pass  a  given 
fixed  point,  be  positive,  and  let  these  con- 
ditions be  indicated  by  the  small  arrows. 
Then,  the  phase  relations  of  the  armature 
electromotive  forces  will  be  represented 
as  in  Fig.  67,  in  which  £3 
lags  behind  E2,  and  E2  lags 
behind  Er  The  potential 
differences  across  the  various 
line  wires  may  then  be  de- 
termined by  vectorial  addi- 
tion and  subtraction;  for 
example,  the  E.M.F.  across 
AB  is  equal  to  the  vectorial 
difference  of  El  and  £2,  since 
they  are  oppositely  directed. 
Taking  the  momentary  posi- 
tive flow  as  directed  towards  A,  then 


Fig.  67. 


Similarly 


EAB  =  #!  e  E2  and  leads  £t  by  30°. 

EBC  =  E2  e  E3  and  lags  90°  behind  Ev 
ECA  =  £3  e  El  and  lags  210°  behind  Er 


ALTERNATORS. 


105 


Calling  the  E.M.F.  generated  in  each  armature  E  volts  as 
before,  the  magnitudes  of  EAB,  EBC,  and  ECA  will  each  be 
\/3  E,  as  may  readily  be  proven  by  geometry.  As  the 
current  flowing  in  each  line  wire  is  the  same  as  that  in  each 
armature,  it  will  be  I  amperes,  and  if  the  circuits  are 
balanced,  i.e.  if  three  loads,  each  having  the  same  resist- 
ance and  the  same  reactance,  are  connected  respectively 
between  A  and  B,  B  and  C,  and  C  and  A,  the  phases  of  the 
currents  in  them  will  be  120°  apart,  as  shown.  Therefore, 
in  a  three-phase,  F-connected  system,  the  voltage  between 
any  two  line  wires  is  Vj  E  volts,  and  the  current  in  each 
line  is  7  amperes. 

Now  let  these  three  alternators  be  connected  as  in  Fig. 
68,    thus"    forming    a    three-phase    mesh-   or    A-connected 


i  \ 

/         ^ 


Fig.  68. 

system.  The  E.M.F.  across  two  line  wires  is  produced  by 
one  alternator  only  and  is  therefore  E  volts,  and  if  all  the 
other  conditions  remain  unchanged,  EBC  will  lag  120° 
behind  EAB)  and  ECA  will  lag  120°  behind  EBC.  Assuming 
the  three  phases  to  be  equally  loaded,  and  representing 
positive  current  flow  in  the  coils  as  their  axes  successively 
pass  a  fixed  point  by  the  small  arrows,  the  magnitudes  of 


io6 


ALTERNATING-CURRENT    MACHINES. 


the  currents  in  the  lines  may  be  determined  vectorially  as 
in  Fig.  68,  where  <£  is  the  angle  of  lag.  Hence 

I  A  =  1 1  ©  ^3  and  lags  30°  +  (j>  behind  Ev 
IB  =  /2  e  /t  and  lags  150°  +  $  behind  Elt 
and        Ic  =  73  e  72  and  leads  El  by  90°  —  </>, 

the  magnitude  of  each  being  V$  I.  Then,  to  sum  up,  the 
voltage  between  any  two  lines  in  a  balanced  three-phase, 
A-connected  system  is  E  volts,  and  the  current  in  each  line 
wire  is  \/3  /  amperes. 

The  power  delivered  by  a  three-phase  alternator  is 
independent  of  the  manner  of  connection,  for  in  one  case 
each  leg  is  supplied  with  /  amperes  at  \/3  E  volts,  and 
in  the  other  case  with  'X/j  /  amperes  at  E  volts. 

46.  Voltage  and  Current  Relations  in  Four-Phase  Systems. 
—  To  obtain  the  current  and  voltage  relations  in  four- 

-A  EDA 


Fig.  69. 


phase  systems,  consider  the  four-phase  alternator  to  con- 
sist  of  four  single-phase  alternators  whose  E.M.F.'s  are 


ALTERNATORS. 


ID/ 


maintained  ninety  degrees  apart  successively.  When  these 
alternators  are  star-connected  as  in  Fig.  69,  it  will  become 
evident  from  an  inspection  of  the  vector  diagram  that  the 
voltages  between  line  wires  are  as  follows,  the  order  of  the 
subscripts  denoting  momentary  positive  direction: 

EAB  =  E^  e  E2  =  V^E  and  leads  El  by  45°, 
EBC  =  E2  e  E3  =  \/2  E  and  lags  45°  behind  E19 
ECD  =  £3  e  E4  =  V^E  and  lags  135°  behind  Elt 
EDA  =  E4  e  El  =  \/2  E  and  lags  225°  behind  Ev 
EAC  =  El  e  E3  =  2  E  and  is  in  phase  with  Ev 
EBD  =  E2  e  E4  =  2  E  and  lags  90°  behind  Er 
If  the  circuits  are  balanced,  the  current  in  each  line  wire  is 
the  same  as  that  flowing  in  an  armature  winding,  or  / 
amperes. 

When  the  four  single-phase  alternators  are  ring-connected 
as  in  Fig.  70,  the  voltage  across  adjacent_line  wires  is  E 
volts,  and  across  alternate  line  wires  is  \/2  E  volts.  The 

current  in  each  line  wire  is  V '2  I  amperes,  r -? A 

and  the  phases  of  these  currents  are  rep- 
resented in  Fig.  71. 

The  relations  of  the  voltages  and  cur- 
rents in  the  armature  windings  of  a  six- 
phase  alternator  to  the  voltages  across 
the  line  wires  and  to  the  currents  therein 
may  similarly  be  determined. 

47.  Measurement  of  Power.  —  The 
power  delivered  to  the  receiving  circuits 
of  a  two-phase,  four-wire  system  can  be 
measured  by  two  wattmeters,  one  con-  Fig.  70. 

nected  in  each  phase.  The  sum  of  their  readings  is  the 
total  power  supplied.  If  the  load  is  balanced,  one  of  the 


io8 


ALTERNATING-CURRENT   MACHINES. 


wattmeters  may  be  dispensed  with,  and  the  total  power  is 
then  double  the  reading  of  the  other. 

In  any  two-phase,  three- 
wire  system  the  power  can 
5  measured  by  two  watt- 
meters connected  as  in  Fig. 
72.  The  sum  of  the  instru- 
ment readings  is  the  whole 
power.  In  a  two-phase, 
three-wire  system,  where  all 
the  load  is  connected  be- 
tween the  outside  wires  and 
the  common  wire,  and  none 
between  the  outside  wires  themselves,  and  where  the  load 
is  balanced,  then  one  wattmeter  can  be  used  to  measure 
the  whole  power  by  connecting  its  current  coil  in  the 
common  wire  and  its  pressure-coil  between  the  common 
wire  and  one  outside  wire  first,  then  shifting  this  con- 
nection to  the  other 
outside  wire,  as  in- 
dicated in  Fig.  73. 
The  sum  of  the  in- 
strument readings  in 
the  two  positions  is 
the  whole  power. 
A  wattmeter  made 
with  two  pressure- 
coils  could  have  one  connected  each  way,  and  the  instru- 
ment would  automatically  add  the  readings,  giving  the 
whole  power  directly.  Or,  again,  a  high  non-reactive  resis- 
tance could  be  placed  between  the  two  outside  wires 
and  the  pressure-coil  of  the  wattmeter  connected  between 


fm-                       f 

4  i 
<^^ 

Load. 

. 

n 

JL 

^fifi?KH^?r^ 

5  , 

Fig.  72. 


ALTERNATORS. 


109 


Fig-  73. 


the  common  wire  and  the  center  point  of  this  resistance. 
This  requires  that  the  wattmeter  be  recalibrated  with  half 
of  this  high  resistance  in  series  with  its  pressure-coil. 

With  the  exception  of  the  two-phase  systems,  the  power 
in  any  balanced  polyphase  system  may  be  measured  by  one 
wattmeter  whose  current  coil  is  placed  in  one  wire,  and 
whose  pressure-coil  is  connected  between  that  wire  and  the 
neutral  point.  The 
instrument  reading 
multiplied  by  the 
number  of  phases 
gives  the  whole  power. 
The  neutral  point 
may  be  on  an  extra 
wire,  as  in  a  three-phase,  four-wire  system;  or  may  be 
artificially  constructed  by  connecting  the  ends  of  equal 
non-reactive  resistances  together,  and  connecting  the  free 
ends  one  to  each  of  the  phase  wires. 

With  the  exception  of  the  two-phase  systems,  the  power 
in  any  w-phase,  w-wire  system,  irrespective  of  balance,  may 
be  determined  by  the  use  of  n  —  i  wattmeters.  The 
current  coils  are  connected,  one  each,  in  n  —  i  of  the  wires, 
and  the  pressure-coils  have  one  of  their  ends  connected  to 
the  respective  phase  wires,  and  their  free  ends  all  connected 
to  the  nih  wire.  The  algebraic  sum  of  the  readings  is  the 
power  in  the  whole  circuit.  Depending  upon  the  power 
factor  of  the  circuit,  some  of  the  wattmeters  will  read 
negatively,  hence  care  must  be  taken  that  all  connections 
are  made  in  the  same  sense;  then  those  instruments  which 
require  that  their  connections  be  changed,  to  make  them 
deflect  properly,  are  the  ones  to  whose  readings  a  negative 
sign  must  be  affixed. 


no 


ALTERNATING-CURRENT   MACHINES. 


Some  specific  connections  for  indicating  wattmeters  in 
three-phase  circuits  are  shown  in  the  following  figures. 
Fig.  74  shows  the  connection  of  three  wattmeters  to  meas- 
ure the  power  in  an  unbalanced  three-phase  system.  All 


Fig.  74- 

the  readings  will  be  in  the  positive  direction,  and  their 
sum  is  the  total  power.  If  a  fourth,  or  neutral  wire  be 
present,  it  should  be  used,  instead  of  creating  an  artificial 
neutral,  as  shown.  The  magnitude  of  the  equal  non- 
reactive  resistances,  used  to  secure  this  neutral  point, 
must  be  so  chosen  that  the  resistances  of  the  pressure-coils 
of  the  wattmeters  will  be  so  large,  compared  thereto,  as 
not  to  disturb  the  potential  of  the  artificial  neutral  point. 

Fig.  75  shows  the  con- 
nection of  one  wattmeter, 
so  as    to   read    one-third 
of   the    whole    power   in 
a  balanced,   three-phase,   || 
four- wire  system.     If  the   ""* 
system    be    three-wire,  a 
neutral    point     may     be 
created  as  in  Fig.  74. 

Another  method  of  measuring  power  in  a  balanced  three- 
phase  system,  either  A-connected  or  F-connected,  is  based 
upon  the  assumption  that  both  pressures  and  currents  vary 


1  —  I    I  — 

Balanced 
Load. 

i   1 

(^=* 

u 

Fig.  75- 


ALTERNATORS. 


Ill 


harmonically.  No  neutral  point  is  required,  and  the  con- 
nections are  shown  in  Fig.  76.  The  free  end  of  the  pressure- 
coil  is  connected  first  to  one  of  the  wires  other  than  that  in 
which  the  current 
coil  is  connected, 
and  then  to  the  other. 
The  angular  dis- 
placements between 
the  current  in  any 
line  wire  and  the 
E.M.F.'s  between 


Fig.  75. 


it  and  the  other  line  wires  are  30°  +  (/>  and  30°  -  <£,  .as 
will  become  evident  from  an  inspection  of  Fig.  67  and 
Fig.  68.  The  readings  of  the  wattmeters  are  then 

Pl  =  V3E/cos(30°  -f  <£) 
and  P2  =  A/3  El  cos  (30°  -  0) 

where  E  is  the  E.M.F.  generated  and  /  is  the  current  flowing 
in  each  armature  coil.  The  algebraic  sum  of  the  read- 
ings is 


_ 

A/3  JE/[jA/3  cos<£—  Jsin^  +  i  A/3  cos  <£  +  J  sin  <£] 
=  3  El  cos  (j> 

which  is  the  total  power  delivered.  When  <j>  is  greater  than 
60°,  Pl  becomes  negative,  hence  care  is  required  to  avoid 
confusion  of  signs  at  low  power  factors.  Both  readings 
will  be  positive  if  the  power  factor  is  greater  than  .5,  but  one 
of  them  will  be  negative  if  it  is  less  than  this  value. 

The  algebraic  difference  of  the  two  wattmeter  readings  is 
PI-  PI  =  ^EI  [cos  (30°  -  <£)  -  cos  (30°  +  <f>)] 

=  v/3_E/[i\/3  cos<£  +  isin<£-J\/3  cos  <£  +  J  sin  <£] 
=  A/3  El  sin  0. 


112 


ALTERNATING-CURRENT   MACHINES. 


It  is  more  convenient,  however,  to  consider  line  voltages 
and  line  currents  instead  of  those  in  the  alternator  armature 
windings  or  in  the  load  of  each  phase.  Therefore,  repre- 
senting the  E.M.F.  between  any  two  line  wires  by  Eh  and 
the  current  in  each  line  by_7/,  then,  since  either  Et  —  v^  E 
(F-connection)  or  It  =  Vj  7  (A-connection),  by  dividing 
the  previous  results  by  \/3,  there  is  obtained 

P\  -P\-  E,I,  sin  4. 

In  the  balanced  three-phase  system  under  consideration, 
it  is  possible  to  determine  the  power  factor  of  the  similar 
receiving  circuits  by  the  use  of  a  single  wattmeter  connected 
as  in  Fig.  76.  The  readings  of  the  wattmeter  in  the  two 
positions  are  the  only  observations  required.  The  power 
factor  is  clearly 

cos  d>  =  cos  tan"1 


[—  [   j— 

Load. 

V  V 

which  is  derived  from  the  two  preceding  equations. 

An  accurate  method  for  the  determination  of  the  power 

in  unbalanced  three- 
phase  systems,  avoid- 
ing the  necessity  of  a 
neutral  point,  involves 
the  use  of  two  watt- 
meters connected  as 
in  Fig.  77.  The  alge- 
braic sum  of  the  in- 
Flg<  77'  strument  indications 

is  the  total  power  supplied.  It  is  possible  to  obtain 
negative  readings,  but  since  the  currents  lag  behind  their 
respective  E.M.F.'s  by  different  amounts  in  an  unbal- 


ALTERNATORS.  113 

anced  system,  it  cannot  be  said  that  when  the  power 
factor  is  less  than  0.5  one  instrument  reads  negatively,  for 
the  term  power  factor  here  has  no  definite  significance. 
To  determine,  then,  the  correct  signs  of  the  wattmeter 
readings,  the  given  load  may  be  replaced  by  a  non-inductive 
balanced  load  of  lamps,  and  if  the  terminals  of  the  potential 
coil  of  one  instrument  must  now  be  reversed  to  deflect  prop- 
erly, it  shows  that  the  negative  sign  must  be  affixed  to  its 
reading  on  the  load  to  be  measured. 

48.  Saturation.  —  The  electromotive  force  produced  in 
an  alternator  at  no-load  is  dependent  upon  the  peripheral 
speed  of  the  rotating  member  and  upon  the  field  excitation. 
The  relation  of  the  open  circuit  voltage  to  the  field  current 
when  the  alternator  is  driven  at  constant  speed  may  be 
represented  by  a  curve  called  the  no-load  saturation  curve. 
For  a  certain  65  K.W.  two-phase  24oo-volt  6o-cycle  inductor 
alternator,  running  at  900  revolutions  per  minute  (air-gap 
of  85  mils),  the  no-load  saturation  curve  has  been  experi- 
mentally determined,  and  is  shown  in  Fig.  78,  curve  A.  It 
indicates,  for  example,  that  the  field  current  necessary  to 
produce  the  rated  voltage  on  open  circuit  when  the  machine 
runs  at  its  proper  speed  is  4.45  amperes.  It  is  seen  that  this 
curve  is  almost  straight  for  small  exciting  currents.  At 
small  excitation,  the  reluctance  of  the  air-gap  is  very  high 
and  that  of  the  iron  very  low,  and  therefore  the  former  may 
be  considered  as  constituting  the  entire  reluctance  of  the 
magnetic  circuit.  Since  the  reluctivity  of  air  is  constant 
regardless  of  the  flux  density,  at  small  excitations  the  flux 
will  be  proportional  to  the  magnetomotive  force,  and  there- 
fore the  open-circuit  voltage  is  proportional  to  the  field 
current,  hence  the  curve  is  straight.  As  the  field  becomes 


ALTERNATING-CURRENT    MACHINES. 


stronger,  however,  the  proportion  of  the  air-gap  reluctance 
to  the  entire  reluctance  decreases,  for  the  permeability  of 
iron  decreases  with  increased  flux-density,  and  therefore 
the  E.M.F.  increases  less  rapidly  with  increased  excitation. 


7 


7 


12          3456-789 

FIELD  AMPERES 

Fig.  78. 

The  percentage  of  saturation  of  an  alternator  at  any 
excitation  may  be  found  from  its  saturation  curve  by  draw- 
ing to  it  a  tangent  at  the  assigned  excitation  and  deter- 
mining its  intercept  on  the  axis  of  ordinates.  The  ratio  of 
this  intercept  to  the  ordinate  of  the  curve  at  the  assigned 
excitation,  expressed  as  a  percentage,  is  the  percentage  of 
saturation.  For  example,  the  percentage  of  saturation  of 
the  alternator  mentioned,  when  the  field  current  is  4.45 


amperes,  is 


95° 
2400 


X  ioo  =39.6%. 


ALTERNATORS.  115 

The  ratio  of  a  small  percentage  increment  of  field  excita- 
tion in  an  alternator  to  the  corresponding  percentage  incre- 
ment of  terminal  voltage  produced  thereby,  is  called  the 
saturation  factor.  Unless  otherwise  specified,  it  refers  to 
the  excitation  existing  at  normal  rated  speed  and  voltage 
and  on  open  circuit.  The  saturation  factor  is  a  criterion 
of  the  degree  of  saturation  and  may  be  expressed  as 


where  m  is  the  percentage  of  saturation.  Thus,  the  satura- 
tion factor  of  the  alternator  whose  saturation  curve  is  shown 
in  Fig.  78,  is 

=  1.66. 

i  -  .396 

The  relation  of  the  terminal  voltage  to  the  field  current 
when  the  alternator  is  driven  at  its  rated  speed  and  deliver- 
ing its  rated  current  is  given  by  the  full-load  saturation  curve, 
which  is  somewhat  similar  in  shape  to  the  no-load  satura- 
tion curve.  It  may  be  determined  experimentally  by 
employing  variable  non-inductive  resistances  for  maintain- 
ing the  constant  full-load  current  on  each  phase,  and  noting 
the  terminal  voltage  corresponding  to  various  excitations. 
The  full-load  saturation  curve  for  the  65  K.W.  inductor 
alternator  at  unity  power  factor  is  shown  as  curve  B  in  Fig. 
78.  As  this  curve  takes  into  account  all  of  the  diverse 
causes  of  decrease  in  terminal  voltage  resulting  from  the 
application  of  a  load  to  the  machine,  it  is  important  in  the 
calculation  of  regulation.  In  alternators  of  large  capacity, 
it  is  a  difficult  matter  to  determine  the  full-load  saturation 
curve  by  test,  and  consequently  other  methods  are  usually 
employed. 


Il6  ALTERNATING-CURRENT    MACHINES. 

If  the  alternator  is  normally  excited  to  above  the  knee  of 
the  saturation  curve,  it  will  require  a  considerable  increase 
of  field  current  to  maintain  the  terminal  voltage  when  the 
load  is  thrown  on,  while,  if  normally  excited  below  the  knee, 
a  slight  increase  of  excitation  will  suffice. 

49.  Regulation.  —  The  regulation  of  an  alternator  is  the 
ratio  of  the  maximum  difference  of  terminal  voltage  from 
the  rated  load  value,  occurring  within  the  range  from  open 
circuit  to  rated  load,  to  the  rated  load  terminal  voltage, 
the  speed  and  field  current  remaining  constant.  As  the 
maximum  deviation  during  this  range  generally  occurs  at 
the  rated  load,  it  is  customary  to  define  regulation  as  the 
ratio  of  the  rise  in  terminal  voltage,  that  occurs  when  full 
load  at  unity  power  factor  is  thrown  off,  to  the  terminal 
voltage.  Or,  expressing  it  in  the  form  of  an  equation, 

_>       ,    .  No-load  Voltage  —  Full-load  Voltage 

Regulation  =  —    — — = — TTTTT  ' 

Full-load  Voltage 

An  alternator  having  perfect  regulation  is  one  which  shows 
no  increase  in  terminal  voltage  upon  opening  its  load  circuit, 
that  is,  the  regulation  is  zero. 

With  small  machines,  the  regulation  can  be  easily  deter- 
mined by  test,  provided  artificial  loads  are  available.  It  is 
simply  necessary  to  plot  the  no-load  and  full-load  saturation 
curves,  and  from  them  the  regulation  at  any  load  can  be 
found.  Referring  to  Fig.  78,  for  example,  the  regulation  of 
the  alternator  at  full  load  with  unity  power  factor  is 

-" —  =.146  or  14.6  per  cent.     With  large  machines, 

2400 

however,  artificial  loads  are  not  usually  available,  and  the 
determination  of  regulation  in  this  case  is  more  difficult  and 
less  accurate. 


ALTERNATORS,  1 1/ 

The  factors  affecting  the  voltage  drop  in  an  alternator 
upon  the  application  of  load  thereto,  are,  the  armature 
resistance,  armature  reactance,  and  magnetization  or 
demagnetization,  occurring  especially  at  low  power  factors. 
These  factors  are  sometimes  grouped  together  and  dealt 
with  collectively  by  the  use  of  a  quantity  called  the  syn- 
chronous impedance.  It  is  that  impedance,  which,  if 
connected  in  series  with  the  outside  circuit  and  to  an 
impressed  voltage  of  the  same  value  as  the  open-circuit 
voltage  at  the  given  speed  and  excitation,  would  permit  a 
current  of  the  same  value  to  flow  as  does  flow. 

The  armature  resistance  drop,  seldom  exceeding  three 
per  cent  of  the  terminal  voltage,  is,  for  each  phase,  equal 
to  the  product  of  the  resistance  of  the  armature  winding  of 
that  phase  and  the  current  flowing  through  it.  In  calcu- 
lating regulation,  the  hot  resistance  (at  75°  C.)  of  the 
windings  is  always  taken. 

The  armature  conductors  of  an  alternator  cut  the  mag- 
netic flux  due  to  the  field  current,  and  this  flux  may  be 
considered  as  sinusoidally  distributed  at  no-load.  An 
E.M.F.  will  thereby  be  produced,  which  will  cause  a  current 
to  flow  through  the  armature  windings  and  through  the 
load  circuit.  The  armature  ampere-turns  set  up  a  mag- 
netic flux  which  is  superimposed  upon  the  field  flux.  The 
magnitude  and  phase  of  the  terminal  electromotive  force 
will  depend  upon  this  resulting  flux,  and,  if  that  due  to  the 
field  excitation  be  constant,  then  the  terminal  E.M.F.  will 
vary  in  a  manner  depending  upon  the  flux  due  to  the  arma- 
ture current,  which,  for  brevity,  will  be  called  the  armature 
flux.  The  armature  self-induction,  being  proportional  to 
the  armature  flux,  varies  and  depends  upon  the  relative 
positions  of  the  armature  and  the  field  and  upon  the  mag- 


ALTERNATING-CURRENT   MACHINES. 


nitude  and  phase  of  the  currents  in  the  armature  windings. 
This  variation  is  shown  in  Fig.  79,  which  gives  the  induc- 
tance corresponding  to  different  angular  positions  of  the 


100       120         110       160       180 


ELECTRICAL  DEGREES 

Fig.  79. 

armature,  zero  degrees  representing  coincidence  of  pole  and 
coil  group  center  lines.  These  curves  refer  to  the  65  K.W. 
two-phase  inductor  alternator  (§  48),  with  a  current  of  9 
amperes  flowing  through  the  winding  of  one  phase  only. 
The  upper  curve  embodies  results  taken  with  the  field  coil 
open-circuited,  and  the  lower  one  with  the  field  coil  short- 
circuited.  Thus,  in  single-phase  alternators,  the  armature 
flux  varies  in  time  and  in  space. 

Consider  a  polyphase  alternator  having  a  revolving 
armature,  a  distributed  armature  winding,  a  magnetic 
circuit  yielding  a  uniform  magnetic  reluctance  as  regards 
the  flux  due  to  the  current  in  any  armature  conductor,  and 
a  balanced  load.  The  armature  flux  will  be  approximately 
sinusoidally  distributed  in  space  and  stationary  as  regards 
the  field  winding,  for  it  would  revolve  backward  as  fast  as 
the  armature  revolves  forward.  The  axis  of  the  armature 
flux,  when  the  current  through  the  conductors  is  in  phase 
with  the  E.M.F.,  is  at  right  angles  to  that  of  the  field  flux, 
as  shown  in  Fig.  80  by  the  dotted  line  sn.  When  the  load 
on  each  phase  is  inductive,  the  axis  of  the  armature  flux  is 
displaced  in  the  direction  of  rotation;  and  when  the  current 


ALTERNATORS. 


Fig.  80. 


supplied  to  the  load  leads  the  E.M.F.  of  the  alternator,  the 
axis  of  the  armature  flux  is  displaced  in  the  direction  oppo- 
site rotation.  These  con- 
ditions are  represented 
by  the  dotted  lines  s'n' 
and  s"n"  respectively. 
From  an  inspection  of 
the  figure,  it  becomes 
evident,  that  with  a  non- 
reactive  load  the  arma- 
ture flux  neither  assists 
nor  opposes  the  field 
flux;  with  an  inductive 
load,  the  armature  flux  has  a  component  which  is  opposite 
to  the  field  flux;  and  with  a  capacity  load  the  armature  flux 
has  a  component  which  is  in  the  same  direction  as  the 
field  flux.  The  magnetomotive  force  causing  the  armature 
flux  may  then  be  considered  as  composed  of  two  compo- 
nents, a  transverse  component,  which  is  a  measure  of  the 
armature  inductance,  and  the  magnetizing  or  demagnetiz- 
ing component,  which  acts  either  with  or  against  the  field 
magnetomotive  force,  depending  upon  the  nature  of  the 
load. 

Commercial  alternators  do  not  have  a  uniform  magnetic 
reluctance,  a  perfectly  distributed  winding,  nor  a  sinusoidal 
flux-distribution;  and  therefore  an  exact  theoretical  treat- 
ment of  alternator  regulation  is  impossible. 

50.  E.M.F.  and  M.M.F.  Methods  of  Calculating  Regu- 
lation. —  Two  methods  of  calculating  the  regulation  of 
alternators  from  the  results  of  other  than  full-load  tests 
have  been  widely  employed,  but  the  results  are  only  approx- 
imate. The  first,  called  the  E.M.F.  method,  generally 


120          ALTERNATING-CURRENT    MACHINES. 

gives  a  larger  regulation,  and  the  second,  called  the  M.M.F. 
or  A.I.E.E.  method,  generally  gives  a  smaller  regulation 
than  what  is  obtained  by  test. 

The  E.M.F.  method  may  be  stated  as  follows:  To 
determine  the  regulation  of  an  alternator  when  supplying 
a  given  current  to  a  receiving  circuit  of  unity  power  factor, 
add  the  armature  resistance  drop  to  the  rated  terminal 
voltage,  and  add  the  sum  vectorially  at  right  angles  to 
the  armature  impedance  voltage,  that  is,  the  open-circuit 
voltage  corresponding  to  the  given  short-circuit  current 
value.  This  result  minus  the  rated  voltage  gives  the 
voltage  rise  at  the  required  load,  and  dividing  this  by  the 
rated  voltage,  the  regulation  at  that  load  will  be  obtained. 

Consider  these  factors  in  detail.  Instead  of  expressing 
the  armature  resistance  drop  in  terms  of  the  resistance  of 
each  winding  and  the  current  therein,  it  is  desirable  in 
practice  to  express  it  in  terms  of  the  line  current  and  the 
resistance  between  any  two  armature  terminals.  For 
example,  take  a  three-phase  alternator  and  assume  it  to  be 
connected  to  a  balanced  load.  Representing  the  line 
voltages  and  line  currents  respectively  by  E  and  /,  and  the 
resistance  of  each  armature  winding  by  r,  then,  in  a  Y- 
connected  alternator,  the  total  copper  loss  is  3  72r,  and  in  a 

A-connected  machine  the  total  copper  loss  is  3(^7=]  r  =  ^r- 
If  R  is  the  armature  resistance  between  terminals,  then 

I  2 

R  =  2  r  in  a  F-connected  alternator,  and  R  =  -     —  =  -  r 

l  +  JL      3 
r       2r 

in  a  A-connected  alternator.  Hence  the  total  copper  loss, 
whether  the  machine  is  Y-  or  A-connected,  is  ~  PR  or  3  P  - 

2  2  . 

To  reduce  this  result  to  an  equivalent  single-phase  circuit 


ALTERNATORS.  121 

with  the  same  voltage  between  line  wires  and  representing 
the  same  power,  P,  consider  that  the  rated  current  per 

P  P 

terminal  in  a  single-phaser  is  — ,  in  a  two-phaser  is ,  and 

E  2  E 

p 
in  a  three-phaser  is  — =»•— .    Denoting  the  equivalent  single- 

v  3  E 

phase  current  by  Ieq,  it  follows  that  in  a  single-phase  cir- 
cuit Ieq  —  7,  in  a  two-phase  circuit  Ieq  =  2  7,  and  in  a 
three-phase  circuit  Ieq  =  \/$  I.  The  equivalent  single-phase 
copper  loss  in  a  three-phase  alternator  is  %J2eqR.  Dividing 
by  Ieq,  there  results  the  equivalent  single-phase  armature 
resistance  drop  of  a  three-phase  alternator,  which  is  |  IeqR. 
This  result  is  also  true  for  two-phase  star-  or  ring-connected 
alternators,  as  may  readily  be  proven.  Thus,  the  armature 
resistance  drop  in  a  polyphase  alternator  is  the  product  of 
the  equivalent  single-phase  current  and  half  the  armature 
resistance  as  measured  between  terminals. 

The  armature  impedance  voltage  is  obtained  from  the 
short-circuit  current  and  the  no-load  saturation  curves. 
The  short-circuit  current  curve  represents  the  field  excita- 
tions required  to  send  various  currents  through  the  short- 
circuited  armature  windings,  and  may  be  obtained  by  direct 
test  without  requiring  large  power  expenditures.  The  short- 
circuit  current  curve  for  the  alternator  considered  in  the 
two  preceding  articles  is  shown  in  Fig.  81. 

As  a  numerical  example,  let  it  be  required  to  determine 
the  regulation  of  this  65  K.w.  two-phase  24oo-volt  alter- 
nator at  full  load  with  unity  power  factor,  the  armature 
resistance  between  terminals  being  5  ohms  at  25°  C. 

The   rated    current    per   terminal   of   the    alternator   is 

P             6^,000 
—JT  =:  — r. =  13.5    amperes,    and   the    equivalent 

p 

single-phase   current  is   —   =27    amperes.     Half    of  the 


122          ALTERNATING-CURRENT   MACHINES. 

armature  resistance  measured  between  terminals  at  75°  C.  is 
$  +  $o  X  2.$  X  .004  =  3  ohms.  Hence  the  armature  IR 
drop  1327X3  =  81  volts  and  is  in  phase  with  the  terminal 
voltage,  since  the  power  factor  of  the  load  is  assumed  to  be 
unity.  The  sum  of  the  armature  resistance  drop  and  the 


ARMATURE  CURRENT  PER  TERMINAL 
S  8  8  £ 

^ 

x^ 

^ 

**^ 

X^ 

X 

,x 

X 

^ 

X 

9123456-789 

FIELD  CURRENT 
Fig.  81. 

terminal  voltage  is  2481  volts.  The  excitation  required  to 
produce  the  rated  current  (13.5  amperes)  is  2.55  amperes, 
as  obtained  from  Fig.  81.  From  the  no-load  saturation 
curve  of  Fig.  78  is  found  the  impedance  voltage  correspond- 
ing to  this  excitation,  and  is  1550  volts.  Adding  the  2481 
volts  and  the  1550  volts  at  right  angles,  there  results  the 
voltage  that  is  considered  from  the  standpoint  of  this  method 
to  be  actually  generated  in  the  alternator, 


V(248i)2  +  (i55o)2  =  2924  volts. 
Hence  the  regulation  at  full  load  with  unity  power  factor  is 

2024  —  2400  ofrr 

v  * — -  =  .218  or  21.8%. 

2400 

The  result  for  the  same  conditions  obtained  from  the  no- 
load  and  the  full-load  saturation  curves  is  14.6%,  thus 
showing  that  the  E.M.F.  method  gives  a  poorer  regulation 
than  is  obtained  by  test. 

Let  it  be  required  to  calculate  the  regulation  of  the  same 


ALTERNATORS. 


123 


alternator  at  f  full-load  by  the  E.M.F.  method,  when  the 
power  factor  of  the  receiving  circuits  is  80%. 

The  rated  j  full-load  current  is  10.125  amperes  and  the 
equivalent  single-phase  current  is  20.25  amperes,  hence  the 
armature  IR  drop  is  20.25  X  3  =  60.75  volts.  The  ter- 
minal E.M.F.  can  be  resolved  into  two  components,  one  in 
phase  with  and  the  other 
at  right  angles  to  the  cur- 
rent. These  components 
are  respectively  2400  X  .8 
or  1920  volts,  and  2400  X 
sin  cos"1. 8  =  2400  X  .6  = 
1440  volts.  The  impedance 
voltage  as  obtained  from 
the  curves  of  Figs.  81  and 
78  is  found  to  be  1220 
volts.  The  result  of  add- 
ing these  E.M.F.'s  in  their  proper  phases  is  the  voltage 
actually  generated  in  the  alternator,  namely 


60.75  )2  +  (1440  +  i22o)2=  3320  volts,   * 
as  shown  diagrammatically  in  Fig.   82.     The  regulation, 
then,  at  J  full-load  and  80%  power  factor  is 


38-3%. 


3320  —  2400 

2400 

The  M.M.F.  method  of  calculating  alternator  regulation 
at  unity  power  factor  may  be  stated  as  follows :  —  The 
exciting  ampere-turns  corresponding  to  the  terminal  voltage 
plus  the  armature  resistance  drop,  and  the  ampere-turns 
corresponding  to  the  impedance  voltage,  are  combined 
vectorially  to  obtain  the  resultant  ampere  turns,  and  the 
corresponding  internal  E.M.F.  is  obtained  from  the  no-load 
saturation  curve.  The  difference  between  this  E.M.F. 


124         ALTERNATING-CURRENT    MACHINES. 

and  the  rated  voltage  is  divided  by  the  rated  voltage  to 
obtain  the  regulation. 

As  a  numerical  example,  let  it  be  required  to  calculate 
the  regulation  of  the  same  alternator  at  full  load  with  unity 
power  factor  by  the  M.M.F.  method. 

The  field  current  corresponding  to  2400  +  81  volts  is 
4.7  amperes,  as  obtained  from  the  no-load  saturation  curve 
of  Fig.  78.  The  field  current  corresponding  to  the  impe- 
dance voltage  (1550  volts)  is  found  from  the  same  curve  and 
is  2.55  amperes.  This  value  can  also  be  obtained  directly 
from  Fig.  81;  it  corresponds  to  the  rated  current  (13.5 
amperes).  Then,  adding  4.7  amperes  and  2.55  amperes 
at  right  angles,  there  results  \/4.72  +  2. 5 52  or  5.35  amperes. 
The  no-load  voltage  corresponding  to  this  excitation  is 
2620  volts  Therefore  the  regulation  is 

2620  —  2400  ,  ~ 

=  .0916  or  9.16%. 

2400 

This  result  is  much  smaller  than  that  obtained  by  test 
(14.6%),  that  is,  the  M.M.F.  method  for  these  conditions 
gives  a  better  regulation  than  it  should.  The  mean  value 
of  the  results  obtained  by  the  E.M.F.  and  M.M.F.  methods 
is  15.5%  and  agrees  fairly  well  with  the  experimental  result; 
but  this  is  not  always  true. 

51.  Regulation  for  Constant  Potential.  —  Alternators 
feeding  light  circuits  must  be  closely  regulated  to  give 
satisfactory  service.  The  pressure  can  be  maintained 
constant  in  a  circuit  by  a  series  boosting  transformer,  but 
it  is  generally  considered  better  to  regulate  the  alternator 
by  suitable  alteration  of  the  field  strength. 

The  simplest  method  of  regulating  the  potential  is  to 
have  a  hand-operated  rheostat  in  the  field  circuit  of  the 
alternator,  when  the  latter  is  to  be  excited  from  a  com- 


ALTERNATORS 


125 


mon  source  of  direct  current,  or  in  the  field  circuit  of  the 
exciter,  if  the  alternator  is  provided  with  one.  The 
latter  method  is  generally  employed  in  large  machines, 
since  the  exciter  field  current  is  small,  while  the  alternator 
field  current  may  be  of  considerable  magnitude,  and  would 
give  a  large  PR  loss  if  passed  through  a  rheostat. 

A   second   method  of  regulation   employs   a  composite 
winding  analogous   to  the  compound  windings  of   direct- 


Fig.  83. 

current  generators.  This  consists  of  a  set  of  coils,  one 
on  each  pole.  These  are  connected  in  series,  and  carry 
a  portion  of  the  armature  current  which  has  been  rectified. 
The  rectifier  consists  of  a  commutator,  having  as  many 
segments  as  there  are  field  poles.  The  alternate  segments 
are  connected  together,  forming  two  groups.  The  groups 
are  connected  respectively  with  the  two  ends  of  a  resist- 
ance forming  part  of  the  armature  circuit.  Brushes, 
bearing  upon  the  commutator,  connect  with  the  terminals 
of  the  composite  winding.  The  magnetomotive  force  of 
the  composite  winding  is  used  for  regulation  only,  the 
main  excitation  being  supplied  by  an  ordinary  separately 
excited  field  winding.  The  rectified  current  in  the  com- 
posite coils  is  a  pulsating  unidirectional  current  that 
increases  the  magnetizing  force  in  the  fields  as  the  cur- 
rent in  the  armature  increases.  The  rate  of  increase  is 


126  ALTERNATING-CURRENT   MACHINES. 

determined  by  the  resistance  of  a  shunt  placed  across  the 
brushes.  By  increasing  the  resistance  of  this  shunt,  the 
amount  of  compounding  can  be  increased.  With  such 
an  arrangement  an  alternator  can  be  over-compounded  to 
compensate  for  any  percentage  of  potential  drop  in  the 
distributing  lines.  The  method  here  outlined  is  used  by 
the  General  Electric  Company  in  their  single-phase 
stationary  field  alternators.  The  connections  are  shown 
in  Fig.  83. 

A  third  method  of  regulation  is  employed  by  the  West- 
inghouse  Company  on  their  revolving  armature  alter- 
nators, one  of  which,  a  75  K.W.,  6o~,  single-phase  machine, 
is  shown  in  Fig.  84.  A  composite  winding  is  employed, 
and  the  compensating  coils  are  excited  by  current  from 
a  series  transformer  placed  on  the  spokes  of  the  armature 
spider.  The  primary  of  this  transformer  consists  of  but 
a  few  turns,  and  the  whole  armature  current  is  conducted 
through  it  before  reaching  the  collector  rings.  The  sec- 
ondary of  this  transformer  is  suitably  connected  to  a 
simple  commutator  on  the  extreme  end  of  the  shaft. 
Upon  this  rest  the  brushes  which  are  attached  to  the  ends 
of  the  compensating  coil.  This  commutator  is  subjected 
to  only  moderate  currents  and  low  voltages.  The  current 
in  the  secondary  of  the  transformer,  and  hence  that  in 
the  compensating  coil,  is  proportional  to  the  main  armature 
current.  The  machine  is  wound  for  the  maximum  desir- 
able over-compounding,  and  any  less  compensation  can  be 
secured  by  slightly  shifting  the  commutator  brushes. 
For  there  are  only  as  many  segments  as  poles  ;  and  if  the 
brushes  span  the  insulation  just  when  the  wave  of  current 
in  the  transformer  secondary  is  passing  through  zero, 
then  the  pulsating  direct  current  in  the  compounding  coil 


ALTERNATORS.  I2/ 

is  equal  to  the  effective  value  of  the  alternating  current ; 
but  if  the  brushes  are  at  some  other  position,  the  current 
will  flow  in  the  field  coil  in  one  direction  for  a  portion  of 
the  half  cycle,  and  in  the  other  direction  for  the  remaining 


Fig.    84. 

portion.  A  differential  action,,  therefore,  ensues,  and  the 
effective  value  of  the  compensating  current  is  less  than  it 
was  before. 

In    order   to  produce  a  constant    potential    on   circuits 
having  a  variable  inductance  as  well  as  a  variable  resist- 


128  ALTERNATING-CURRENT   MACHINES. 

ance,  the  General  Electric  Co.  has  designed  its  compensated 
revolving  field  generators,  which  are  constructed  for  two- 
or  three-phase  circuits.  The  machine,  Fig.  85,  is  of  the 


Fig.  85. 

revolving  field  type,  the  field  being  wound  with  but  one 
simple  set  of  coils.  On  the  same  shaft  as  the  field,  and 
close  beside  it,  is  the  armature  of  the  exciter,  as  shown 
in  Fig.  86.  The  outer  casting  contains  the  alternator 
armature  windings,  and  close  beside  them  the  field  of  the 
exciter.  This  latter  has  as  many  poles  as  has  the  field  of 
the  alternator.  Alternator  and  exciter,  therefore,  operate 
in  a  synchronous  relation.  The  armature  of  the  exciter  is 
fitted  with  a  regular  commutator,  which  delivers  direct 
current  both  to  the  exciter  field  and,  through  two  slip- 


ALTERNATORS.  129 

rings,  to  the  alternator  field.  On  the  end  of  the  shaft, 
outside  of  the  bearings,  is  a  set  of  slip-rings,  four  for  a 
quart er-phaser,  three  for  a  three-phaser,  through  which 
the  exciter  armature  receives  alternating  current  from  one 
or  several  series  transformers  inserted  in  the  mains  which 
lead  from  the  alternator.  This  alternating  current  is 
passed  through  the  exciter  armature  in  such  a  manner  as 
to  cause  an  armature  reaction,  as  described  in  §  49,  that 
increases  the  magnetic  flux.  This  raises  the  exciter  vol- 
tage and  hence  increases  the  main  field  current.  The 


Fig.   86. 

reactive  magnetization  produced  in  the  exciter  field  is 
proportional  to  the  magnitude  and  phase  of  the  alternating 
current  in  the  exciter  armature.  The  reactive  demag- 
netization of  the  alternator  field  is  proportional  to  the 
magnitude  and  phase  of  the  current  in  the  alternator 
armature.  And  these  currents  have  the  fixed  relations 
of  current  strength  and  phase,  which  are  determined  by  the 
series  transformers.  Hence  the  exciter  voltage  varies  so 
as  to  compensate  for  any  drop  in  the  terminal  voltage. 
Neither  the  commutator  nor  any  of  the  slip-rings  carry 
pressures  of  over  75  volts.  The  amount  of  over-corn- 


130 


ALTERNATING-CURRENT    MACHINES. 


pounding  is  determined  by  the  ratio  in  the  series  trans- 
formers. The  normal  voltage  of  the  alternator  may  be 
regulated  by  a  small  rheostat  in  the  field  circuit  of  the 
exciter.  The  various  connections  of  this  type  of  com- 
pensated alternator  are  shown  in  Fig.  87. 
The  regulation  of  voltages  by, means  of  composite  wind- 


Fig.  87. 


ings  finds  application  on  alternators  up  to  about  250  K.w. 
output.  The  Tirrill  Regulator,  for  use  with  large  or  small 
generators,  is  made  by  the  General  Electric  Company,  and 
shown  in  Fig.  88.  This  device  operates  by  rapidly  opening 
and  closing  a  shunt  circuit  connected  across  the  exciter 
field  rheostat,  the  operation  being  accomplished  by  means 
of  a  differentially  wound  relay,  which  is  connected  to  the 
exciter  bus-bars.  There  are  two  control  magnets,  one  for 
direct  current  and  the  other  for  alternating  current.  The 
current  for  the  first  is  taken  from  the  exciter  bus-bars,  and 
the  current  for  the  latter  is  taken  from  a  potential  trans- 
former connected  in  the  circuit  to  be  regulated.  Upon 
the  same  spool  as  this  potential  winding  is  an  adjustable 
compensating  winding  which  is  connected  to  the  secondary 


ALTERNATORS.  131 

of  a  current  transformer  inserted  in  the  principal  lighting 
circuit.  The  cores  of  these  control  magnets  are  attached 
to  pivoted  levers  provided  with  contacts  at  their  other  ends. 


Fig.  88. 


When  a  load  is  thrown  on  the  alternator,  the  voltage  will 
tend  to  drop  and  the  alternating-current  magnet  will 
weaken,  thus  causing  the  main  contacts  to  close.  This 


132 


ALTERNATING-CURRENT   MACHINES. 


also  causes  the  relay  contacts  to  close  and  short-circuit  the 
exciter    field    rheostat,    thereby    increasing    the    potential 


Fig.   89. 


supplied  to  the  alternator  field.  The  general  scheme  and 
connections  of  this  regulator  for  a  single  generator  and 
exciter  are  shown  in  Fig.  89. 


ALTERNATORS.  133 

52.  Efficiency.  —  The  following  is  abstracted  from  the 
Report  of  the  Committee  on  Standardization  of  the  Ameri- 
can Institute  of  Electrical  Engineers.  Only  those  por- 
tions are  given  which  bear  upon  the  efficiency  of  alternators. 
They  will,  however,  apply  equally  well  to  synchronous 
motors. 

The  "efficiency"  of  an  apparatus  is  the  ratio  of  its  net 
power  output  to  its  gross  power  input. 

Electric  power  should  be  measured  at  the  terminals  of 
the  apparatus. 

In  determining  the  efficiency  of  alternating-current 
apparatus,  the  electric  power  should  be  measured  when 
the  current  is  in  phase  with  the  E.M.F.  unless  otherwise 
specified,  except  when  a  definite  phase  difference  is  in- 
herent in  the  apparatus,  as  in  induction  motors,  etc. 

Where  a  machine  has  auxiliary  apparatus,  such  as  an 
exciter,  the  power  lost  in  the  auxiliary  apparatus  should 
not  be  charged  to  the  machine,  but  to  the  plant  consisting 
of  the  machine  and  auxiliary  apparatus  taken  together. 
The  plant  efficiency  in  such  cases  should  be  distinguished 
from  the  machine  efficiency. 

The  efficiency  may  be  determined  by  measuring  all  the 
losses  individually,  and  adding  their  sum  to  the  output  to 
derive  the  input,  or  subtracting  their  sum  from  the  input 
to  derive  the  output.  All  losses  should  be  measured  at, 
or  reduced  to,  the  temperature  assumed  in  continuous 
operation,  or  in  operation  under  conditions  specified. 

In  synchronous  machines  the  output  or  input  should  be 
measured  with  the  current  in  phase  with  the  terminal 
E.M.F.  except  when  otherwise  expressly  specified. 

Owing  to  the  uncertainty  necessarily  involved  in  the 
approximation  of  load  losses,  it  is  preferable,  whenever 


134  ALTERNATING-CURRENT    MACHINES. 

possible,  to  determine  the  efficiency  of  synchronous  ma- 
chines by  input  and  output  tests. 

The  losses  in  synchronous  machines  are : 

a.  Bearing  friction  and  windage. 

b.  Molecular    magnetic    friction    and    eddy  currents  in 
iron,  copper,  and  other  metallic  parts.     These  losses  should 
be  determined  at  open  circuit  of  the  machine  at  the  rated 
speed  and  at    the  rated  voltage,  +  IR  in  a  synchronous 
generator,  —  IR  in  a  synchronous  motor,  where  /  =  cur- 
rent in  armature,  R  =  armature  resistance.     It  is  undesir- 
able to  compute  these  losses  from  observations  made  at 
other  speeds  or  voltages. 

These  losses  may  be  determined  by  either  driving  the 
machine  by  a  motor,  or  by  running  it  as  a  synchronous 
motor,  and  adjusting  its  fields  so  as  to  get  minimum  cur- 
rent input,  and  measuring  the  input  by  wattmeter.  The 
former  is  the  preferable  method,  and  in  polyphase  ma- 
chines the  latter  method  is  liable  to  give  erroneous  results 
in  consequence  of  unequal  distribution  of  currents  in  the 
different  circuits  caused  by  inequalities  of  the  impedance 
of  connecting  leads,  etc. 

c.  Armature-resistance    loss,  which   may  be    expressed 
by  p  I^R  ;  where  R  =  resistance  of  one  armature  circuit 
or  branch,  /  =  the  current    in    such  armature    circuit   or 
branch,    and  /  =  the    number    of    armature     circuits    or 
branches. 

d.  Load  losses.     While    these    losses    cannot   well    be 
determined   individually,   they  may  be   considerable,   and, 
therefore,  their  joint  influence  should    be  determined  by 
observation.     This  can  be  done  by  operating  the  machine 
on  short  circuit  and  at  full-load  current,  that  is,  by  deter- 
mining what  may  be  called  the  "short-circuit  core  loss." 


ALTERNATORS. 


135 


With  the  low  field  intensity  and  great  lag  of  current 
existing  in  this  case,  the  load  losses  are  usually  greatly 
exaggerated. 

One-third  of  the  short-circuit  core  loss  may,  as  an 
approximation,  and  in  the  absence  of  more  accurate  infor- 
mation, be  assumed  as  the  load  loss. 

e.  Collector-ring  friction  and  contact  resistance.  These 
are  generally  negligible,  except  in  machines  of  extremely 
low  voltage. 

/.  Field  excitation.  In  separately  excited  machines, 
the  PR  of  the  field  coils 
proper  should  be  used. 
In  self-exciting  machines, 
however,  the  loss  in  the 
field  rheostat  should  be 
included. 

The  efficiency  curve  of 
an  alternator  may  be 
plotted  when  the  losses  at 
different  loads  have  been 
determined.  The  efficiency 
curve  of  a  5000  K.  w. 
11,000  volt  alternator, 
and  that  of  a  1000  K.  w. 
500  volt  alternator  are 
shown  in  Fig.  90. 

U0         20         40         GO          80        100        120 
PERCENT  FULL-LOAD 

53.  Rating.  —  Alterna- 
tors are  rated  by  their 

electrical  output,  either  in  kilowatts  or  in  kilovolt-amperes. 
By  rating  is  meant  the  power  that  the  machine  can  deliver 
to  the  load  without  an  excessive  rise  in  temperature.  This 


100 

s^ 

^ 

^ 

10001 

•^7" 

80 

t 

/ 

^"  70 

I 

v 

30 

0 

136  ALTERNATING-CURRENT   MACHINES. 

temperature  rise  is  due  to  the  losses  in  the  alternator;  these 
include  the  practically  constant  iron  losses  and  the  copper 
losses,  variable  with  load.  Under  a  fixed  current  output 
the  temperature  of  the  armature  will  rise  until  the  rate  of 
escape  of  heat  from  it  is  equal  to  the  rate  of  its  develop- 
ment. This  ultimate  temperature  should  not  exceed  80°  C. 
in  any  case.  Under  a  given  voltage  the  current  output  is 
limited  by  the  rise  of  temperature  and  the  power  output 
is  further  limited  by  the  power  factor  of  the  load  circuit. 
Hence  the  power  supplied  by  an  alternator  to  a  reactive 
load  is  less  than  that  supplied  to  a  non-reactive  load  for 
the  same  temperature  rise  in  the  machine.  It  is  advisable 
to  rate  alternators  in  kilovolt-amperes  and  to  specify  the 
power  factor  on  which  this  rating  is  based.  Thus,  a  6600 
volt  alternator  whose  rated  current  is  500  amperes,  called 
a  3300  kilovolt-ampere  alternator,  could  deliver  3300  kilo- 
watts to  a  non-reactive  load,  but,  for  the  same  temperature 
rise  it  could  only  deliver  2640  kilowatts  to  a  load  of  80% 
power  factor. 

An  alternator  should  be  able  to  carry  a  25%  overload  for 
two  hours  without  serious  injury  because  of  heating,  elec- 
trical or  mechanical  stresses,  and  with  an  additional  tem- 
perature rise  not  exceeding  15°  C.  above  that  specified  for 
rated  load,  the  overload  being  applied  after  the  machine 
has  acquired  the  temperature  corresponding  to  continuous 
operation  at  rated  load. 

54.  Inductor  Alternators.  —  Generators  in  which  both 
armature  and  field  coils  are  stationary  are  called  inductor 
alternators.  Fig.  91  shows  the  principle  of  operation  of 
these  machines.  A  moving  member,  carrying  no  wire, 
has  pairs  of  soft  iron  projections,  which  are  called  indue- 


ALTERNATORS. 


137 


tors.  These  projections  are  magnetized  by  the  current 
flowing  in  the  annular  field  coil  as  shown  in  figure.  The 
surrounding  frame  has  internal  projections  corresponding 


/ARMATURE  COILS  ^ 


Fig.  91. 

to  the  inductors  in  number  and  size.  These  latter  projec- 
tions constitute  the  cores  of  armature  coils.  When  the 
faces  of  the  inductors  are  directly  opposite  to  the  faces  of 
the  armature  poles,  the  magnetic  reluctance  is  a  minimum, 
and  the  flux  through  the  armature  coil  accordingly  a  maxi- 
mum. For  the  opposite  reason,  when  the  inductors  are 
in  an  intermediate  position  the  flux  linked  with  the  arma- 
ture coils  is  a  minimum.  As  the  inductors  revolve,  the 
linked  flux  changes  from  a  maximum  to  a  minimum,  but  it 
does  not  change  in  sign. 

Absence  of  moving  wire  and  the  consequent  liability  to 
chafing  of  insulation,  absence  of  collecting  devices  and 
their  attendant  brush  friction,  and  increased  facilities  for 
insulation  are  claimed  as  advantages  for  this  type  of  ma- 
chine. By  suitably  disposing  of  the  coils,  inductor  alter- 
nators may  be  wound  for  single-  or  polyphase  currents. 

The  Stanley  Electric  Manufacturing  Company  manu- 
factured two-phase  inductor  alternators.  A  view  of  one  of 
their  machines  is  given  in  Fig.  92,  with  the  frame  separated 
for  inspection  of  the  windings.  In  this  picture  the  field 


138 


ALTERNATING-CURRENT    MACHINES. 


coil  is  hanging  loosely  between  the  pairs  of  inductors.     The 
theoretical  operation  of  this   machine  is   essentially   that 


Fig.  92. 

described    above.     All    iron    parts,    both    stationary    and 

revolving,    that   are   subjected   to   pulsations   of   magnetic 

flux,  are  made  up  of 

laminated  iron.  The 

large    field     coil    is 

wound  on  a  copper 

spool.         Ordinarily 

when  the  field  circuit 

of  a  large  generator 

is  broken,  theE.M.F. 

of  self-induction  may 

rise    to    so    high    a 

value  as  to  pierce  the  Fig.  93 

insulation.    With  this  construction  the  copper  spool  acts  as 

a  short  circuit  around  the  decaying  flux,  and  prevents  high 


ALTERNATORS. 


139 


£.M.F.'s  of  self-induction.     Figs.  93  and  94  show  details 
of  construction  of  larger  machines  of  this  type. 

55.    Revolving  Field  Alternators.  —  In  this  type  of  al- 
ternator, the  armature  windings  are  placed  on  the  inside 


140  ALTERNATING-CURRENT    MACHINES. 

of  the  surrounding  frame,  and  the  field  poles  project  radi- 
ally from  the  rotating  member.     As  was  stated  before   this 


type  of  construction  is  to  be  recommended  in  the  case  of 
large  machines  which  are  required  to  give  either  high 
voltages  or  large  currents.  With  the  same  peripheral 


ALTERNATORS. 


141 


velocity,  there  is  more  space  for  the  armature  coils;  the 
coils  can  be  better  ventilated,  air  being  forced  through 
ducts  by  the  rotating  field;  stationary  coils  can  be  more 
perfectly  insulated  than  moving  ones;  and  the  only  cur- 
rents to  be  collected  by  brushes  and  collector  rings  are 
those  necessary  to  excite  the  fields. 

Fig.  95  shows  a  General  Electric  750  K.  w.  revolving 
field  generator.  The  two  collector  rings  for  the  field  cur- 
rent are  shown,  and  in  Fig.  96  the  edgewise  method  of 


Fig.  96. 

winding  the  field  coils  is  shown.  The  collector  rings  are 
of  cast  iron  and  the  brushes  are  of  carbon.  Fig.  97  shows 
the  details  of  construction  of  a  5000  K.W.  three-phase 
66oo-volt  machine  of  this  type  as  constructed  for  the 
Metropolitan  Street  Railway  Co.  of  New  York.  This 
machine  has  40  poles,  runs  at  75  R.P.M.  at  a  peripheral 
velocity  of  3900  feet  per  minute.  This  gives  a  frequency 
of  25.  The  air  gap  varies  from  five-sixteenths  at  the 
pole  center  to  eleven-sixteenths  at  the  tips.  The  short- 
circuit  current  at  full-load  excitation  is  less  than  800  am- 


142 


ALTERNATING-CURRENT    MACHINES. 


peres  per  leg.     The  rated  full-load  current  is  slightly  over 
300  amperes. 

Fig.  98  shows  the  method  of  assembling  the  armature 


6CALE  f*  INCH  EQUl,lJS  ONE  FOOT 


Fig.  97- 


coils  in  the  slots  of  the  stationary  core.     In  this  machine 
there  is  a  three-phase  winding  distributed  so  as  to  utilize 


ALTERNATORS. 


Fig. 98. 


144  ALTERNATING-CURRENT   MACHINES. 


Fig.  99- 


two  slots  per  pole  per  phase.  Fig.  99  shows  the  construction 
of  a  rotating  field  which  consists  of  a  steel  rim  mounted 
upon  a  cast-iron  spider.  Into  dovetailed  slots  in  the  rim  are 
fitted  laminated  plates  with  staggered  joints.  These  plates 


ALTERNATORS.  145 

are  bolted  together.  The  laminations  are  supplied  at 
intervals  with  ventilating  ducts.  The  coils  are  kept  in 
place  by  retaining  wedges  of  non-magnetic  material. 

56.  Self-Exciting  Alternators.  —  An  alternator  of  a  dif- 
ferent type  from  those  previously  considered  is  the  Alex- 
anderson  self-exciting  alternator  which  is  manufactured 
by  the  General  Electric  Company.  The  armature  and 
field  windings  differ  in  no  respect  from  the  usual  type  used 
in  alternating-current  generators.  The  field  current  is 


derived  from  an  auxiliary  winding  placed  in  the  same  slots 
as  the  main  armature  winding,  and  is  .rectified  by  means  of 
a  segmental  commutator  with  one  active  segment  per  pole. 
The  revolving  field  of  a  100  K.W.  three-phase  self-exciting 
alternator  with  its  commutator  is  shown  in  Fig.  100.  Alter- 
nate segments  of  this  commutator  are  connected  to  two 
steel  rings  surrounding  the  segments,  and  these  rings  are 
connected  to  the  field  winding.  A  terminal  of  each  aux- 
iliary winding  is  connected  to  one  of  three  brushes  bearing 
on  this  commutator,  and  their  other  terminals  are  attached 
to  a  three-phase  rheostat,  as  shown  in  Fig.  101. 


146 


ALTERNATING-CURRENT    MACHINES. 


Automatic  compounding  is  effected  by  series  trans- 
formers connected  as  indicated  in  the  figure.  The  amount 
of  boosting  in  the  field  circuit  will  depend  upon  the  values 


Fig.  101. 

of  the  currents  in  the  secondaries  of  the  series  transformers 
as  well  as  upon  the  power  factor  of  the  load. 


PROBLEMS. 

1.  The  armature  of  a  25  cycle,  eight  pole  single-phase  alternator  has 
three  slots  per  pole  with  ten  conductors  in  each  slot,  the  slots  occupying 
one-half  of  the  pole  distance.     If  the  flux  from  each  pole  is  i, 200,000 
maxwells,  what  will  be  the  effective  E.M.F.  generated  in  the  armature, 
assuming  this  E.M.F.  to  be  sinusoidal? 

2.  Determine  the  voltage  across  the  outside  terminals  of  a  two-phase 
three-wire  100  volt  alternator,  when  the  windings  on  its  armature  are 
set  85°  apart  instead  of  90°  apart. 

3.  The  E.M.F.  generated  in  each  armature  winding  of  a  three-phase 
alternator  is  125  volts,  and  the  current  in  each  is  5  amperes  when  the 
alternator  is  connected  to  a  certain  load.     Determine  the  voltages  be- 
tween the  lines  and  the  current  flowing  in  each  line  wire,  when  the 
machine  is  F-connected  and  when  it  is  A-connected. 

4.  Find  the  magnitudes  and  phases  of  the  various  voltages  across  the 


PROBLEMS.  147 

line  wires  of  a  four-phase  star-connected  system,  the  voltage  generated 
in  each  armature  winding  being  75  volts. 

5.  A  three-phase  alternator  is  connected  to  a  balanced  non-reactive 
receiving  circuit.     It  is  required  to  determine  the  magnitude  of  the  power 
supplied  by  the  alternator,  when  the  voltage  across  the  line  wires  is 
150  volts  and  the  current  in  each  line  is  20  amperes. 

6.  A  three-phase  alternator  is  connected  to  a  balanced  inductive  load 
and  the  power  is  measured  according  to  the  method  of  Fig.  76.     What 
is  the  power  factor  of  the  receiving  circuits  if  the  observed  indications 
on  the  wattmeter  are  2,200  and  2,900  watts  respectively? 


ARMATURE  VOLTS 

_ 

!i 

il 

400 
300 

200 
100 

^ 

x--* 

^*" 

x 

^ 

/ 

/ 

, 

/ 

/ 

/ 

// 

l_ 

0          20         40         GO         80         100       120        140       160       180       200 
FIELD  AMPERES 

Fig.  102. 

7.  In  the  alternator  of  Fig.  78,  determine  the  saturation  factor  when 
the  exciting  current  is  six  amperes. 

8.  Calculate  the  regulation  of  the  alternator  of  the  preceding  problem 
when  the  field  excitation  is  7  amperes,  and  when  the  power  factor  of  the 
load  is  unity. 

9.  The  no-load  saturation  and  the  armature  short-circuit  current 
curves  of  a  3500  K.  w.  three-phase  6,600  volt  revolving  field  alternator 
are  shown  in  Fig.  102.    'Calculate  the  regulation  at  full  load  with  unity 


148  ALTERNATING-CURRENT    MACHINES. 

power  factor  by  the  E.M.F.  method.     The  armature  resistance  between 
terminals  is  .093  ohms  at  25°  C. 

10.  Determine  the  regulation  of  the  alternator  of  the  preceding 
problem  on  an  inductive  load  of  80%  power  factor,  by  the  E.M.F. 
method. 

11.  If  the  load  of  the  preceding  problem  were  replaced  by  a  capacity 
load  of  the  came  power  factor,  what  would  be  the  regulation  at  full  load, 
as  calculated  by  the  E.M.F.  method. 

12.  Determine  the  regulation  of  the  alternator  of  Fig.  102  for  each  of 
the  conditions  of  the  three  preceding  problems,  applying  the  M.M.F. 
method. 


THE   TRANSFORMER.  149 


CHAPTER   VI. 

THE   TRANSFORMER. 

57.  Definitions.  —  The  alternating-current  transformer 
consists  of  one  magnetic  circuit  interlinked  with  two  elec- 
tric circuits,  of  which  one,  the  primary,  receives  electrical 
energy,  and  the  other,  the  secondary,  delivers  electrical 
energy.  If  the  electric  circuits  surround  the  magnetic 
circuit,  as  in  Fig.  103,  the  transformer  is  said  to  be  of  the 
core  type.  If  the  re- 
verse is  true,  as  in 
Fig.  104,  the  trans- 
former is  of  the  shell 
type.  The  practical 
utility  of  the  trans- 
former lies  in  the  fact 
that,  when  suitably 
designed,  its  primary 
can  take  electric  energy 
at  one  potential,  and 
its  secondary  deliver 
the  same  energy  at  Fig.  103. 

some  other  potential;  the  ratio  of  the  current  in  the  primary 
to  that  in  the  secondary  being  approximately  inversely 
as  the  ratio  of  the  pressure  on  the  primary  to  that  on 
the  secondary. 

The  ratio  of  transformation  of  a  transformer  is  repre- 


ISO          ALTERNATING-CURRENT    MACHINES. 

sented  by  r,  and  is  the  ratio  of  the  number  of  turns  in  the 
secondary  coils  to  the  number  of  turns  in  the  primary  coil. 
This  would  also  be  the  ratio  of  the  secondary  voltage  to 


Fig.  104. 

the  primary  voltage  if  there  were  no  losses  in  the  trans- 
former. A  transformer  in  which  this  ratio  is  greater  than 
unity  is  called  a  " step-up"  transformer,  since  it  delivers 
electrical  energy  at  a  higher  pressure  than  that  at  which 
it  is  received.  When  the  ratio  is  less  than  unity  it  is 
called  a  "step-down"  transformer.  Step-up  transformers 
find  their  chief  use  in  generating  plants,  where  because  of 
the  practical  limitations  of  alternators,  the  alternating  cur- 
rent generated  is  not  of  as  high  a  potential  as  is  demanded 
for  economical  transmission.  Step-down  transformers  find 
their  greatest  use  at  or  near  the  points  of  consumption  of 
energy,  where  the  pressure  is  reduced  to  a  degree  suitable 
for  the  service  it  must  perform.  The  conventional  repre- 
sentation of  a  transformer  is  given  in  Fig.  105.  In  general, 
little  or  no  effort  is  made  to  indicate  the  ratio  of  trans- 
formation by  the  relative  number  of  angles  or  loops  shown, 


THE    TRANSFORMER.  151 

though   the   low-tension   side   is    sometimes    distinguished 
from  the  high-tension  side  by  this  means. 

When  using  the  same  or  part  of  the  same  electric  cir- 
cuit for  both  primary  and  secondary,  the  device  is  called 
an  auto-transformer.  These  are  sometimes  used  in  the 


Fig.  105. 

starting  devices  for  induction  motors,  and  sometimes 
connected  in  series  in  an  alternating-current  circuit,  and 
arranged  to  vary  the  E.M.F.  in  that  circuit.  Fig.  106  is 
the  conventional  representation  of  an  auto-transformer. 

58.  The  Ideal  Transformer.  —  The  term  ideal  transformer 
may  be  applied  to  one  possessing  neither  hysteresis  and 
eddy  current  losses  in  the  core  nor  ohmic  resistance  in  the 
windings,  and  all  the  flux  set  up  by  one  coil  links  with  the 
other  coil  also.  Actual  transformers,  however,  do  not 
satisfy  these  conditions,  yet  their  behavior  approximates 
closely  to  that  of  an  ideal  transformer. 

When  the  secondary  coil  of  a  transformer  is  open-circuited 
it  is  perfectly  idle,  having  no  influence  on  the  rest  of  the 
apparatus,  and  the  primary  becomes  then  merely  a  choke 
coil  or  reactor.  The  reactance  of  a  commercial  trans- 
former is  very  large  and  its  resistance  very  small,  con- 
sequently the  impedance  is  high  and  almost  wholly  reactive. 
In  the  ideal  transformer  the  current  that  will  flow  in  the 
primary  when  the  secondary  is  open-circuited  is  very  small 
and  lags  90°  behind  the  E.M.F.  which  sends  it.  This 
current  is  called  the  exciting  current,  and  will  be  sinusoidal 
in  the  ideal  transformer  when  the  impressed  electromotive 


152  ALTERNATING-CURRENT   MACHINES. 

force  is  sinusoidal.  A  flux  will  be  set  up  in  the  iron  of  the 
transformer,  which  is  sinusoidal  and  in  phase  with  the 
exciting  current.  This  flux  induces  a  sinusoidal  E.M.F. 
in  the  primary  coil  which  is  90°  behind  the  flux  in  phase 
because  the  induced  E.M.F.  is  greatest  when  the  time  rate 
of  flux  change  is  greatest,  and  this  flux  change  is  greatest 
when  passing  through  the  zero  value.  This  induced 
electromotive  force  is  90°  behind  the  flux,  which  in  turn  is 
90°  behind  the  impressed  E.M.F. ;  therefore  the  induced 
E.M.F.  is  180°  behind  the  impressed  electromotive  force, 
or  is  a  counter  E.M.F.  In  the  ideal  transformer  under 
consideration,  the  counter  pressure  is  exactly  equal  to  the 
E.M.F.  impressed  upon  the  primary.  The  phase  relations 
of  the  pressures,  the  exciting  current,  and  the  flux  in  an 
ideal  transformer  are  shown  in  Fig.  107.  It  should  be 
noted  that  the  exciting  current,  being 
at  right  angles  to  the  pressure  for  this 
transformer,  does  not  represent  a  loss 
in  power,  for  the  energy  is  alternately 
,E  received  from  and  supplied  to  the 
>xo  OF  FLUX  circuit.  This  may  be  shown  graphi- 
cally as  in  Fig.  108,  where  the  lobes  of 
negative  and  positive  power  are  equal. 
When  the  secondary  winding  of  an 
ideal  transformer  is  closed  through  an 

Fig.  107.  m      m   & 

outside  impedance,  the  variations  in 
the  flux,  which  is  linked  with  the  secondary  as  well  as 
the  primary,  produce  in  the  secondary  an  E.M.F.  r  times 
as  great  as  the  counter  E.M.F.  in  the  primary,  since  there 
are  r  times  as  many  turns  in  the  secondary  coil  as  there 
are  in  the  primary,  or 

Es  =  rEp; 


THE   TRANSFORMER. 


153 


and  a  current  /,  will  flow  through  the  external  circuit.  The 
ampere-turns  of  the  secondary,  nsls,  will  be  opposed  to  the 
ampere-turns  of  the  primary,  and  will  thus  tend  to  demag- 
netize the  core.  This  tendency  is  opposed  by  a  readjust- 
ment of  the  conditions  in  the  primary  circuit.  Any  demag- 
netization tends  to  lessen  the  counter  E.M.F.  in  the  primary 
coil,  which  immediately  allows  more  current  to  flow  in  the 
primary,  and  thus  restores  the  magnetization  to  a  value  but 
slightly  less  than  the  value  on  open-circuited  secondary. 
Thus  the  core  flux  remains  practically  constant  whether 


Fig.  108. 

the  secondary  be  loaded  or  not,  the  ampere  turns  of  the 
secondary  being  opposed  by  a  but  slightly  greater  number 
of  ampere  turns  in  the  primary.  So 

nsls  =  nplp,  very  nearly, 
and  7, 


The  counter  E.M.F.  in  the  primary  of  a  transformer 
accommodates  itself  to  variations  of  load  on  the  secondary 


154 


ALTERNATING-CURRENT   MACHINES. 


in  a  manner  similar  to  the  variation  of  the  counter  E.M.F. 
of  a  shunt  wound  motor  under  varying  mechanical  loads. 

The  vector  diagram  of  the  ideal  transformer,  when  the 
secondary  is  closed  through  a  circuit  having  a  reactance 
X2,  and  a  resistance  R2,  is  shown  in  Fig.  109.  It  represents 
a  step-down  transformer  where  T  =  }.  The  secondary 

=       "1  —  2 


current  7,  lags  behind  Es  by  the  angle  $  =  tan" 


The 


primary  ampere-turns  are  composed  of  two  components  — 
one  necessary  to  balance  the  secondary  ampere-turns  and 
the  other  necessary  to  magnetize  the  core,  as  shown. 

When  the  transformer  delivers  little  power,  the  magne- 
tizing component  of  the  magneto- 
motive force  is  comparable  in 
magnitude  with  the  active  com- 
*  ponent,  consequently  the  secon- 
dary current  lags  behind  the 
primary  current  by  less  than  180°. 
When  the  transformer  is  fully 
loaded,  however,  the  magnetizing 
current  is  comparatively  small 
and  therefore  the  directions  of 
Ip  and  Is  are  very  nearly  oppo- 
site. This  is  also  true  of  com- 
mercial transformers,  in  which  the  exciting  current  is  less 
than  90°  behind  Ep. 

59.  Core  Flux.  —  The  relation  between  the  magnetic 
flux  in  a  transformer  core  and  the  primary  impressed 
E.M.F.  can  be  determined  by  considering  that  the  flux 
varies  harmonically,  and  that  its  maximum  value  is  <£m; 
then  the  flux  at  any  time,  /,  is  3>m  cos  cot,  and  the  counter 


Fig.  109. 


THE    TRANSFORMER. 


155 


EM.  P.,   which  is   equal   and  opposed   to   the  impressed 
primary  pressure  Ep,  may  be  written  (§  13,  vol.  i.) 

77  /  _  «£.  d(®mcosa>t) 
" 


and  since  &m  and  CD  are  constant 


from  which 


and 


Epm  =  io~ 


dt 


sin  w/, 


nptu 


This  equation  is  used  in  designing  transformers  and 
choke  coils.  The  values  of  <E>W  for  60  cycle  transformers 
of  different  capacities,  as  determined  by  experiment  and 
use,  are  shown  in  the  curve,  Fig.  no.  It  is  usual  in  such 


Total'  Flux  in  Megamaxwells  . 

»  *  -.  i  s  c  :  • 

_  

^^^- 

^ 

-^ 

^ 

^ 

x 

^ 

/ 

Lighting  Transformers 

/ 

/ 

/ 

10        12        14         16        18      20        22        24 

Capacity  in  Kilowatts 

Fig.   no. 


designs  to  assume  a  maximum  flux  density,  (Bw.  Trans- 
former cores  are  worked  at  low  flux  densities,  and,  while 
the  value  assumed  differs  considerably  with  the  various 
manufacturers,  it  is  safe  to  say  that  for  25  cycles  (Bw  varies 


156  ALTERNATING-CURRENT  MACHINES. 

between  6  and  8  kilogausses;  for  60  cycles  between  5  and  6 
kilogausses;  and  for  125  cycles  between  3  and  4  kilogausses. 
The  necessary  cross-section,  A,  of  iron  in  square  centimeters 
is  found  from  the  relation 


60.  Transformer  Losses.  —  The  transformer  as  thus  far 
discussed  would  have  100  %  efficiency,  no  power  whatever 
being   consumed   in   the   apparatus.     The   efficiencies   of 
loaded  commercial  transformers  are  very  high,  being  gen- 
erally above  95  %  and  frequently  above  98  %.     The  losses 
in  the  apparatus  are  due  to  the  resistance  of  the  electric 
circuits,  hysteresis,  and  eddy  currents.     These  losses  may 
be  divided  into  core  losses  and  copper  losses,  according  as  to 
whether  they  occur  in  the  iron  or  the  wire  of  the  trans- 
former. 

61.  Core  Losses.  —  (#)   Eddy  current  loss.     If  the  core 
of  a  transformer  were  made  of  solid  iron,  strong  eddy  cur- 
rents would  be  induced  in  it.     These  currents  would  not 
only  cause  excessive  heating  of  the  core,  but  would  tend 
to  demagnetize  it,  and  would  require  excessive  currents  to 
flow  in  the  primary  winding  in  order  to  set  up  sufficient 
counter  E.M.F. 

To  a  great  extent  these  troubles  are  prevented  by  mak- 
ing the  core  of  laminated  iron,  the  laminae  being  trans- 
verse to  the  direction  of  flow  of  the  eddy  currents  but 
longitudinal  with  the  magnetic  flux.  Each  lamina  is  more 
or  less  thoroughly  insulated  from  its  neighbors  by  the 
natural  oxide  on  the  surface  or  by  Japan  lacquer.  The 
eddy  current  loss  is  practically  independent  of  the  load. 

An  empirical  formula  for  the  calculation  of  the  watts 


THE   TRANSFORMER. 


157 


lost  in  the  transformer  core  due  to  eddy  currents,  based 
upon  the  assumption  that  the  laminations  are  perfectly 
insulated  from  one  another,  is 


Pe   = 


where 


k  =  a   constant    depending    upon   the    resistivity  of 

the  iron, 

v  =  volume  of  iron  in  cm.3, 
/  =  thickness  of  one  lamina  in  cm., 
/  =  frequency, 
and 

<&m  =  maximum  flux  density  ($m  per  cm.2). 

In  practice  k  has  a  value  of  about  1.6  X  io~n. 

The  values  of  Pe  in  watts  per  cubic  inch  and  per  pound 
in  terms  of  flux  density  for  25  cycle  and  60  cycle  transformers 
may  be  taken  directly  from  the  curves  of  Fig.  in.  These 


3456789       10 
KILO-MAXWELLS  PER  SQ.  CM. 
Fig.  in. 

curves  are  plotted  from  values  calculated  by  means  of  the 
formula  and  refer  to  laminations  usually  employed,  these 
being  0.014  incn  thick. 

(b)   Hysteresis  loss.     A  certain  amount  of  power,  Ph, 
due  to  the  presence  of  hysteresis,  is  required  to  carry  the 


I58 


ALTERNATING-CURRENT   MACHINES. 


iron  through  its  cyclic  changes.     The  value  of  Ph  can  be 
calculated  from  the  formula  expressing  Steinmetz's  Law, 


Ph  =  icr* 


where 


v  =  volume  of  iron  in  cm.3, 
/  =  frequency, 

(Bm  =  the  maximum  flux  density, 
and          T)  =  the  hysteretic  constant. 

A  fair  value  of  rj  for  transformer  sheets  is  .0021.  Curves 
of  the  hysteresis  loss  in  25  cycle  and  60  cycle  transformers 
based  upon  this  value  of  rj  are  shown  in  Fig.  112.  In  the 
better  grades  of  transformers,  however,  the  hysteresis  loss 


3456789       10 

KILO-MAXWELLS  PER  SQ.  CM, 

Fig.  112. 

is  less  than  that  indicated  by  the  curves  by  about  15  %. 
Hysteresis  loss  is  practically  independent  of  the  load. 

In  modern  commercial  transformers  the  core  loss  at  60^ 
may  be  about  75  %  hysteresis  and  25  %  eddy  current  loss. 
At  125  ^  it  may  be  about  60  %  hysteresis  and  40  %  eddy 
current  loss.  This  might  be  expected,  since  it  was  shown 
that  the  first  power  of  /  enters  into  the  formula  for  hysteresis 
loss,  while  the  second  power  of  /  enters  into  the  formula 
for  eddy  current  loss. 


THE   TRANSFORMER.  159 

The  core  loss  is  also  dependent  upon  the  wave-form  of 
the  impressed  E.M.F.,  a  peaked  wave  giving  a  somewhat 
lower  core  loss  than  a  flat  wave.  It  is  not  uncommon  to 
find  alternators  giving  waves  so  peaked  that  transformers 
tested  by  current  from  them  show  from  5  %  to  10  %  less 
core  loss  than  they  would  if  tested  by  a  true  sine  wave. 
On  the  other  hand  generators  sometimes  give  waves  so 
flat  that  the  core  loss  will  be  greater  than  that  obtained  by 
the  use  of  the  sine  wave. 

The  magnitude  of  the  core  loss  depends  also  upon  the 
temperature  of  the  iron.  Both  the  hysteresis  and  eddy  cur- 
rent losses  decrease  slightly  as  the  temperature  of  the  iron 
increases.  In  commercial  transformers,  a  rise  in  tempera- 
ture of  40°  C.  will  decrease  the  core  loss  from  5  %  to  10  %. 
An  accurate  statement  of  the  core  loss  thus  requires  that 
the  conditions  of  temperature  and  wave-shape  be  specified. 

62.  Exciting  Current.  —  In  commercial  transformers  the 
exciting  current  lags  less  than  90°  behind  the  primary  im- 
pressed E.M.F.,  because  of  the  iron  losses.  The  exciting 
current  may  therefore  be  resolved  into  two  components', 
one  in  phase  with  the  primary  E.M.F.,  and  the  other  at 
right  angles  to  it.  The  former  is  that  current  necessary  to 
overcome  the  core  losses  and  is  called  the  power  component 
of  the  exciting  current.  It  is  expressed  as 

,     _  p  .  +  p  » 


the  values  of  Pe  and  Ph  being  calculated  from  the  formulae 
of  §  61. 

The  other  component,  being  90°  behind  Ep,  is  termed  the 
wattless  component  of  the  exciting  current,  or  the  magnetizing 


160  ALTERNATING-CURRENT  MACHINES. 

current  of  a  transformer.  It  is  that  current  which  sets  up  the 
magnetic  flux  in  the  core,  and  is  denoted  by  the  symbol  Imag. 
Representing  the  reluctance  of  the  core  by  CR,  and  the 
magnetomotive  force  necessary  to  produce  the  flux  <£m  by 
OC,  from  §§  21  and  25,  vol.  L, 


_  oc_ 

(R  (R 

whence  Imafl  = 


4  *»p      4  \/2  TTW, 
The  value  of  (R  is  calculated  (§  24,  vol.  i.)  from 


where  /  is  the  length  of  magnetic  circuit,  A  its  cross-section 
and  p  the  reluctivity  of  the  iron 


/         T                i  \ 

i  p  =  —  = j . 

\         A6      permeability/ 


permeability/ 

The  phase  relations  of  the  power  and  wattless  components 
,E  of  the  exciting  current  are  shown 

in  Fig.  113.  The  angle  between 
Iexc  and  Imag  is  called  the  angle  of 
hysteretic  advance  and  is  denoted 
by  a.  This  angle  is  determined 
from  the  relation 


Imag  ' 

Fig>  "3t  It  should  be  noted  that  the  use 

of  the  term  hysteretic  in  this  connection  is  somewhat  mis- 
leading, for  the  value   of  a  depends  upon  the  eddy  cur- 


THE   TRANSFORMER.  l6l 

rent  loss  as  well  as  upon  the  hysteresis  loss.     The  exciting 


current  is 


lexc    —    vPmag    +  Pe+h 

and  lags  behind  the  primary  impressed  E.M.F.  by  an  angle 
90°  -  a. 

The  magnitude  and  position  of  the  exciting  current  of  a 
transformer  can  be  determined  experimentally  by  the  use 
of  a  wattmeter,  a  voltmeter,  and  an  ammeter  connected  in 
the  primary  circuit,  the  secondary,  of  course,  being  open- 
circuited.  The  ammeter  reading  gives  the  value  of  Iexc  and 
its  position  is  given  by  the  equation 


P  being  the  wattmeter  reading  minus  the  copper  loss  due  to 
the  exciting  current  in  the  primary  winding. 

If  the  impressed  E.M.F.  be  harmonic  the  flux  will  also 
be  harmonic  and  consequently  the  magnetizing  current 
cannot  be  harmonic,  because  of  the  variation  in  the  reluc- 
tance of  the  core.  Besides  a  decrease  in  permeability  with 
increasing  flux  density,  the  permeability  on  rising  flux  is 
smaller  than  on  falling,  under  a  given  magnetomotive  force, 
due  to  hysteresis.  Therefore  the  magnetizing  current  wave 
will  be  peaked  and  will  have  a  hump  on  the  rising  side. 
This  magnetizing  current  wave  can  be  plotted  when  the 
hysteresis  loop  of  the  core  is  given  over  the  range  of  flux 
density  produced  by  the  primary  E.M.F.  Since  OC  is 
directly  proportional  to  the  current,  and  (fc  is  proportional 
to  <$,  if  proper  units  are  chosen,  3Cm  may  be  taken  equal 
to  I  mag  m,  and  (BTO  may  be  taken  equal  to  &m.  Then  the 
hysteresis  loop  and  the  sinusoidal  flux  curve  may  be  drawn 


1 62 


ALTERNATING-CURRENT   MACHINES. 


as  in  Fig.  114.  The  value  of  the  magnetizing  current 
corresponding  to  a  given  value  of  the  flux  is  obtained  by 
taking  the  abscissa  corresponding  to  this  flux  value  from 
the  hysteresis  loop  and  laying  it  off  as  an  ordinate  at  the 
point  on  the  time  axis  corresponding  to  the  flux  value  taken. 
This  process  is  indicated  in  the  figure,  and  the  entire  current 
curve  has  been  constructed  by  proceeding  in  this  manner. 

This  distorted  curve  of  magnetizing  current  may  be 
resolved  into  true  sine  components  (§  10),  a  fundamental 
with  higher  harmonics,  the  third  harmonic  being  the  most 
pronounced.  The  exciting  current,  being  composed  of 


Fig.  114. 

two  components,  one  of  which  is  non-sinusoidal,  will  also  be 
non-sinusoidal,  but  since  it  is  usually  very  small  compared 
with  the  load  current,  no  appreciable  error  will  be  intro- 
duced by  considering  Iexc  as  harmonic. 

The  exciting  current  varies  in  magnitude  with  the  design 
of  the  transformer.  In  general  it  will  not  exceed  5  %  of  the 
full  load  current,  and  in  standard  lighting  transformers  it 
may  be  as  low  as  i  %.  In  transformers  designed  with 


THE   TRANSFORMER. 


163 


joints  in  the  magnetic  circuit  the  magnitude  of  the  exciting 
current  is  largely  influenced  by  the  character  of  the  joints, 
being  large  if  the  joints  are  poorly  constructed. 

63.  Equivalent  Resistance  and  Reactance  of  a  Trans- 
former. —  If  a  current  of  definite  magnitude  and  lag  be 
taken  from  the  secondary  of  a  transformer,  a  current  of 
the  same  lag  and  T  times  that  magnitude  will  flow  in  the 
primary,  neglecting  resistance,  reluctance,  and  hysteresis. 
An  impedance  which,  placed  across  the  primary  mains, 
would  allow  an  exactly  similar  current  to  flow  as  this 
primary  current,  is  called  an  equivalent  impedance,  and  its 
components  are  called  equiva- 
lent resistance  and  equivalent 
reactance. 

If  the  secondary  winding  of 
the  transformer  have  a  resis- 
tance Rs  and  a  reactance  Xs, 
and  if  the  load  have  a  resist-  Fig 

ance  R2  and  a  reactance  X2, 
then  the  current  that  will  flow  in  the  secondary  circuit  is 


E. 


/.  = 


where  Es  is  the  secondary  induced  pressure  when  Ep  is  the 
primary  impressed  E.M.F.  The  secondary  current  lags 
behind  Es  by  an  angle  <j>  whose  tangent  is 

X.  +  X2 


For  convenience,   X2  and  R2  will  be  taken  equal  to  zero, 
Fig.  115,  and  the  expressions  which  will  result  will  be  the 


164          ALTERNATING-CURRENT    MACHINES. 

equivalent  resistance  and  reactance  of  the  secondary  wind- 
ing of  the  transformer. 


Therefore  VR?  +  x)  =      -  . 

*s 

If  the  equivalent  impedance  have  a  resistance  R  and  a 

X          X 

reactance  X  then  the  ratios        and  ™  must  be  equal,  since 
R  Rs 

the  angle  of  current  lag  is  the  same  in  both  primary  and 
secondary.  And  since  the  current  in  the  equivalent  im- 
pedance has  the  same  magnitude  as  that  in  the  primary 


and 
But 
and 

therefore, 


B,,t  R        Rs 

x    ¥/ 

Solving  R  =  —  k 


-17-     -1      -y 

-A     —  ~  -A«j 

which   are   the   values   of   the   equivalent   resistance    and 
reactance  of   the  secondary  winding  respectively.      Simi- 


THE   TRANSFORMER.  165 

larly   the   equivalent  load   resistance   and    reactance    are 
respectively 


and  X  =  -  X2. 

64.  Copper  Losses.  —  The  copper  losses  in  a  transformer 
are  almost  solely  due  to  the  regular  current  flowing 
through  the  coils.  Eddy  currents  in  the  conductor  are 
either  negligible  or  considered  together  with  the  eddy  cur- 
rents in  the  core. 

When  the  transformer  has  its  secondary  open-circuited 
the  copper  loss  is  merely  that  due  to  the  exciting  current 
in  the  primary  coil,  PexcRp.  This  is  very  small,  much 
smaller  than  the  core  loss,  for  both  Iexc  and  Rp  are  small 
quantities.  When  the  transformer  is  regularly  loaded  the 
copper  loss  in  watts  may  be  expressed 

pc  =  I;RP  +  I*R., 

where  Rp  and  Rs  are  the  resistances  of  the  primary  and 
secondary  coils  respectively.  In  an  ideal  transformer 
with  zero  reluctance,  Is  is  180°  behind  Ip,  and  this  is  also 
approximately  true  for  a  commercial  transformer  under  a 
considerable  load.  Therefore,  for  convenience,  the  sec- 
ondary resistance  may  be  reduced  to  the  primary  circuit 
and  the  copper  loss  may  then  be  expressed  as 

P.=I>(R,+R)  =  /„'(*,  +  ±R,). 

At  full  load  this  loss  will  considerably  exceed  the  core 
loss.  While  the  core  loss  is  constant  at  all  loads,  the 
copper  loss  varies  as  the  square  of  the  load. 


1 66 


ALTERNATING-CURRENT   MACHINES. 


65.  Efficiency.  —  Since  the  efficiency  of  induction  appa- 
ratus depends  upon  the  wave-shape  of  E.M.F.,  it  should  be 
referred  to  a  sine  wave  of  E.M.F.,  except  where  expressly 
specified  otherwise.  The  efficiency  should  be  measured 
with  non-inductive  load,  and  at  rated  frequency,  except 
where  expressly  specified  otherwise. 

The  efficiency  of  a  transformer  is  expressed  by  the  ratio 
of  the  net  power  output  to  the  gross  power  input  or  by 
the  ratio  of  the  power  output  to  the  power  output  plus  all 
the  losses.  The  efficiency,  e,  may  then  be  written, 


vsis  +  ph+pe  +  p' 

where  Vs  is  the  difference  of  potential  at  the  secondary 
terminals. 

The  losses  and  efficiencies  of  a  line  of  2200  volt,  60  cycle 
transformers  of  the  shell  type  are  given  in  the  following 
table:  — 


Rated  Output   in 
Kilowatts. 

Core  Loss  in  Watts  . 

Full-load  Copper 
Loss  in  Watts. 

Per  cent  Efficiency 
at  Full-load. 

I 

3° 

32 

94.1 

2 

5° 

56 

94.9 

3 

66 

78 

95-4 

5 

90 

i°5 

.       96.3 

7«5 

116 

135 

96.8 

10 

135- 

170 

97.0 

15 

169 

233 

97-4 

20 

200 

3°4 

97-5 

25 

225 

375 

97-65 

3° 

250 

444 

97-8 

40 

300 

586 

97-9 

5° 

350 

725 

98.0 

The   efficiencies   of   a   certain    10   K.W.  transformer   at 
various  loads  are  shown  by  the  curve  of  Fig.  116. 


THE   TRANSFORMER. 


167 


If  the  transformer  be  artificially  cooled,  as  many  of  the 
larger  ones  are,  then  to  this  denominator  must  be  added 
the  power  required  by  the  cooling  device,  as  power  con- 


20       40       60       80      100     120     140 
PER  CENT  FULL  LOAD 
Fig.  116. 

sumed  by  the  blower  in  air-blast  transformers,  and  power 
consumed  by  the  motor-driven  pumps  in  oil  or  water 
cooled  transformers.  Where  the  same  cooling  apparatus 
supplies  a  number  of  transformers  or  is  installed  to  supply 
future  additions,  allowance  should  be  made  therefor. 

Inasmuch  as  the  losses  in  a  transformer  are  affected  by 
the  temperature,  the  efficiency  can  be  accurately  specified 
only  by  reference  to  some  definite  temperature,  such  as 
75°  C. 

The  all-day  efficiency  of  a  transformer  is  the  ratio  of 


168  ALTERNATING-CURRENT   MACHINES. 

energy  output  to  the  energy  input  during  the  twenty-four 
hours.  The  usual  conditions  of  practice  will  be  met  if  the 
calculation  is  based  on  the  assumption  of  five  hours  full- 
load  and  nineteen  hours  no-load  in  transformers  used  for 
ordinary  lighting  service.  With  a  given  limit  to  the  first 
cost,  the  losses  should  be  so  adjusted  as  to  give  a  maximum 
all-day  efficiency.  For  instance,  a  transformer  supplying 
a  private  residence  with  light  will  be  loaded  but  a  few 
hours  each  night.  It  should  have  relatively  much  copper 
and  little  iron.  This  will  make  the  core  losses,  which  con- 
tinue through  the  twenty-four  hours,  small,  and  the  copper 
losses,  which  last  but  a  few  hours,  comparatively  large. 
Too  much  copper  in  a  transformer,  however,  results  in  bad 
regulation.  In  the  case  of  a  transformer  working  all  the 
time  under  load,  there  should  be  a  greater  proportion  of 
iron,  thus  requiring  less  copper  and  giving  less  copper  loss. 
This  is  desirable  in  that  a  loaded  transformer  has  usually 
a  much  greater  copper  loss  than  core  loss,  and  a  halving 
of  the  former  is  profitably  purchased  even  at  the  expense 
of  doubling  the  latter. 

66.   Calculation   of   Equivalent   Leakage   Inductance.  — 

The  magnetic  leakage  in  a  transformer  is  that  flux  which 
links  with  one  winding  and  not  with  the  other.  Its  mag- 
nitude depends  upon  the  size  and  form  of  the  coils  and  the 
manner  of  their  arrangement.  This  magnetic  leakage  may 
be  considered  equivalent  to  an  inductance  connected  in  the 
primary  circuit  and  to  an  inductance  connected  in  the 
secondary  circuit.  After  the  leakage  .flux  has  been  deter- 
mined, the  inductances  Lp  and  Ls  are  found  from  the 
relation 

*=»?,  §12 


THE   TRANSFORMER. 


169 


and  then  the  reactances  are  obtained  from  Xp  =  ajLp  and 
Xs  =  ojLs. 

To  calculate  the  leakage  flux,  consider  a  shell  type  trans- 
former having  one  primary  and  one  secondary  coil  with  many 
turns  of  wire  in  each.  The  paths  of  the  leakage  flux  in 
this  type  of  transformer  are  indicated  in  Fig.  117.  Let  the 


X 


Fig.  117. 

dimensions  shown  on  the  sections  be  expressed  in  centi- 
meters. 

It  is  convenient  to  consider  the  leakage  flux  as  the  sum 
of  three  portions,  the  part  passing  through  the  primary 
space,  the  part  passing  through  the  secondary  space,  and 
the  part  passing  through  the  gap,  g.  The  magnetomotive 
force  tending  to  send  flux  through  the  elementary  portion 

dp  and  back  through  the  iron  is  ~  of  the  whole  M.M.F.  of 
the  primary,  so  for  any  element 


M.M.F.  =  4  7inpip  £ 


I/O          ALTERNATING-CURRENT   MACHINES. 

where  ip  is  expressed  in  absolute  units.  Since  the  per- 
meability of  iron  is  roughly  1000  times  that  of  air,  no 
appreciable  error  is  introduced  by  considering  the  whole 
reluctance  of  the  circuit  of  the  leakage  flux  to  be  in  the  air 
portion  of  that  circuit.  The  cross-section  area  of  this  air 
portion  of  the  magnetic  circuit  for  any  element  is 

-  2  dp  =  Up, 

and  its  length   is  /,  therefore  the  reluctance  is  -  —  .    The 

Up 

elementary  primary  leakage  flux,  d$p,  is  then 
M.M.F. 


IP 


Inductance,  as  denned  in  §  12,  is  numerically  equal  to 
the  number  of  linkages  per  absolute  unit  of  current,  or 

<£ 

l  =  n—>    The  number  of  turns  linked  with  the  elementary 

flux  d®p  is  -^  of  the  total  number  of  primary  turns,  therefore 
the  elementary  leakage  inductance  dlp  is 

M       P  „  <!®p       P   „    4™pivXpdp     4  T 

~  ~~ 


Integrating  over  the  full  width  of  the  primary  coil  from 
o  to  P,  there  is  obtained 

P^ 

Proceeding  in  like  manner,  the  value  of  ls  is  found  to  be 
7      4  nn?X     S 

'•-   i   •  i 


THE   TRANSFORMER.  171 

The  leakage  flux  passing  through  the  gap  between  the 
coils  is  set  up  by  the  entire  magnetomotive  forces  of  either 
the  primary  or  the  secondary.  Consider  the  flux  passing 
through  the  right-hand  half  of  the  gap  to  be  set  up  by  the 
secondary  M.M.F.,  and  that  through  the  other  half  of  the 
gap  to  be  set  up  by  the  primary  M.M.F.  Then,  proceeding 
as  before,  the  leakage  inductances  equivalent  to  these 
portions  of  the  leakage  flux  are  respectively 

A.  Tl'Yl  2/l        J? 

l»~~f~  '   a 
and  -t       **>&.&. 

I  2 

Adding;  the  leakage  inductances  due  to  the  primary  and 
secondary  M.M.F.'s  are  respectively, 

4  KHpX  II 

I       \3 

4  Tin 2  X I S     g  \ 
and  /s+^=L^_fy. 

Reducing  to  practical  units,  and  multiplying  by  a),  the  pri- 
mary and  secondary  leakage  reactances  are  respectively 


and  X 

all  the  dimensions  being  in  centimeters. 

The  secondary  leakage  reactance  may  be  reduced  to  the 
primary  circuit  by  dividing  by  r2  (§  63),  but  it  should  be 
remembered  that  this  is  only  permissible  when  the  trans- 
former is  under  considerable  load  or  when  the  exciting 


1/2 


ALTERNATING-CURRENT    MACHINES. 


current  is  entirely  ignored,  as  in  most  practical  calculations. 
The  total  equivalent  leakage  reactance  in  the  primary 
circuit  is  then 


(3) 


As  some  of  the  leakage  flux  passes  through  and  between 
the  coils  where  they  project  beyond  the  core,  it  is  usual  to 
take  for  A  the  mean  length  of  a  turn  of  a  coil  diminished  by 
|  of  the  length  extending  beyond  the  iron. 

The  minimum  leakage  reactance  would  result  if  each 
secondary  turn  were  immediately  adjacent  to  a  primary 
turn,  but  obviously  this  ideal  condition  cannot  be  attained 
in  practice.  Still  it  may  be  approximated  by  interleaving 
the  secondary  and  primary  coils.  When  one  coil  is  placed 
between  the  two  halves  of  the  other,  as  in  Fig.  141,  the 
leakage  reactance  is  approximately  one  fourth  of  that 
expressed  by  the  foregoing  formulae.  The  values  assigned 
to  the  symbols  for  this  case  are  indicated  in  Fig.  1 18.  Thus, 


Fig.  118. 


by  having  many  coils  and  by  alternating  primary  and 
secondary  coils,  the  leakage  reactance  may  be  greatly 
reduced. 


THE   TRANSFORMER.  173 

The  formulae  for  the  calculation  of  leakage  reactance 
.  may  also  be  applied  to  the  core  type  of  transformers,  but 
the  notation  will  be  slightly  different.  With  this  type, 
P  and  3  are  the  radial  depths  of  the  primary  and  secondary 
coils  respectively,  g  is  the  radial  width  of  the  gap,  I  is  the 
axial  length  of  the  coils,  ^  is  the  mean  length  of  a  turn  of 
the  windings,  and  np  and  ns  are  the. number  of  primary 
and  secondary  turns  respectively  on  both  sections. 

67.  Regulation.  —  The  definition  of  the  regulation  of  a 
transformer  as  recommended  by  the  American  Institute  of 
Electrical  Engineers  is  as  follows:  "In  constant-potential 
transformers,  the  regulation  is  the  ratio  of  the  rise  of  second- 
ary terminal  voltage  from  rated  non-inductive  load  to  no- 
load  (at  constant  primary  impressed  terminal  voltage)  to 
the  secondary  terminal  voltage  at  rated  load."  Further 
conditions  are  that  the  frequency  be  kept  constant,  and 
that  the  wave  of  impressed  E.M.F.  be  sinusoidal. 

Not  the  whole  of  the  primary  impressed  E.M.F.  is 
operative  in  producing  secondary  pressure,  for  IpRp  volts 
are  expended  in  overcoming  the  resistance  of  the  primary 
coil,  and  ISRS  volts  are  expended  in  overcoming  the  resist- 
ance of  the  secondary  coil.  In  addition  to  these,  a  part  of 
the  impressed  E.M.F.  is  lost  in  overcoming  the  primary 
and  secondary  reactances  due  to  the  leakage  flux,  the 
magnitudes  of  these  decrements  being  IPXP  and  ISXS. 
To  consider  these  various  losses  in  voltage,  imagine  the 
transformer  itself  to  be  an  ideal  one,  but  to  have  a  resist- 
ance Rs,  equal  to  the  resistance  of  the  secondary  coil,  and 
a  reactance  XtJ  equal  to  the  secondary  'leakage  reactance 
of  the  actual  transformer,  connected  in  the  secondary 
circuit  in  series  with  the  load  resistance  R2  and  reactance 


174 


ALTERNATING-CURRENT   MACHINES. 


X2.  And  further,  let  there  be  a  resistance  Rp,  equal  to  the 
resistance  of  the  primary  coil  of  the  actual  transformer, 
and  a  reactance  Xp,  equal  to  the  primary  leakage  reactance 
thereof,  connected  in  the  primary  circuit  of  the  ideal  trans- 
former, as  shown  in  Fig.  119. 

The  complete  vector  diagram  of  E.M.F.'s  and  currents  in 


PR.      SEC. 

Fig.  119. 

a  transformer  corresponding  to  the  arrangement  of  Fig.  119 

is  represented  in  Fig.  120,  where 

Vs  —  the  difference  of  potential  at  the  secondary  terminals, 

Es  =  E.M.F.  induced  in  secondary  winding, 

Ep  =  impressed  primary  pressure, 

E0  =  operative  part  of  Ep) 

Ip  and  Is  =  primary  and  secondary  currents  respectively. 

For  clearness  a  i  to  i  ratio  has  been  portrayed,  and  the 

various  drops  are  greatly  exaggerated.     The  diagram  will 

be  discussed  in  detail. 

The  exciting  current,  Iexc,  has  two  components,  namely 
Imag  in  phase  with  the  flux,  and  Ie+h  in  phase  with  £0. 
The  magnetizing  current  is  determined  from  the  expression 

10  l$>m 
Imag    =    -  —  -      —  ,  §02 

4  v  2  xAjj.np 
and  the  power  component  of  the  exciting  current  is  obtained 


from 


161 


THE    TRANSFORMER. 
The  current  flowing  in  the  secondary  circuit  is 


175 


§  62 


and  lags  or  leads  the  secondary  induced  E.M.F.  by  an 
angle  (/>  whose  tangent  is 

X.  +.  X, 
R,+R2' 

the  secondary  induced  elec- 
tromotive force  being  90° 
behind  the  flux.  The  pri- 
mary current,  Ip,  is  equal 
to  the  vectorial  sum  of  —  r 
times  the  secondary  cur- 
rent and  the  exciting  cur- 
rent as  shown.  When  a 
small  current  is  taken  from 
the  secondary  of  the  trans- 
former, the  directions  of  Ip 
and  Is  are  considerably  less 
than  1 80°  apart,  but  when 
the  secondary  current  is 
large,  the  directions  of  Ip 
and  Is  are  approximately 
opposite. 

The  secondary  induced 
E.M.F.  is  not  all  utilizable 
at  the  terminals.  There  is  a  resistance  drop  of  ISRS  volts 
which  is  in  phase  with  7S,  and  a  reactance  drop  of  Is  Xs 
volts  due  to  the  leakage  flux,  this  bein'g  at  right  angles  to 
the  phase  of  the  secondary  current.  The  result  of  subtract- 


Fig.  120. 


1/6  ALTERNATING-CURRENT    MACHINES. 

ing  ISRS  and  ISXS  from  Es  vectorially  is  Vs,  which  is  the 
difference  of  potential  at  the  secondary  terminals. 

The  operative  part,  £0,  of  the  primary  impressed  elec- 
tromotive force  which  is  necessary  to  produce  the  secondary 
induced  pressure  Es,  leads  the  latter  by  180°  and  its  mag- 

£ 
nitude  is  — -.     There  is  a  primary  resistance  drop  of  IPRP 

volts  in  phase  with  Ip  and  a  reactive  drop  due  to  leakage 
of  IPXP  volts  at  right  angles  to  Ip.  Therefore  the  E.M.F. 
impressed  upon  the  primary  terminals  necessary  to  produce 
E0  is  the  vectorial  sum  of  E0,  IPRP  and  Ip  Xp,  and  is  denoted 
byEp. 

Both  Rs  and  Rp  become  known  quantities  as  soon  as 
the  size  of  the  secondary  and  primary  conductors  is 
known.  The  values  of  Xs  and  Xp  are  calculated  from 
the  formulae  derived  in  Art.  66.  Thus  all  the  quantities 
entering  into  the  calculation  of  the  vectors  shown  in  Fig.  120 
are  known. 

Then,  when  Is  is  the  full-load  current,  the  regulation  of 
the  transformer  at  power  factor  =  cos  $  is 

^-7  ;: 

Regulation  =  — - 


which,  when  multiplied  by  100,  gives  the  percentage  regu- 
lation. 

A  circuit  approximately  equal  to  that  of  Fig.  119  is  shown 
as  Fig.  121,  where  the  secondary  resistances  and  reactances 
are  reduced  to  the  primary  circuit,  and  where  the  exciting 
current  is  considered  as  flowing  through  a  separate  impe- 
dance, thus  eliminating  all  transformer  action. 


THE   TRANSFORMER.  177 

A  transformer  diagram  of  practical  importance  is  depend- 
ent upon  the  consideration  that  the  exciting  current  may 
be  neglected  when  the  apparatus  carries  a  large  load.  It 


wwwvm  —  i 


°5 
§1 


Fig.  tax. 

follows  that  Ip  =  rls,  and  that  Ip  is  exactly  opposite  /,. 
The  primary  and  secondary  resistance  drops,  being  in 
phase  respectively  with  Ip  and  Is,  are  parallel,  and  the 
latter  may  be  reduced  to  the  primary  circuit  and  added 
algebraically  to  IPRP.  Then  the  total  equivalent  resistance 

/               7?  \ 
drop  of  the  transformer  is  Ip  (  Rp  -\ |l.     Similarly  the 

total  equivalent  reactance  drop  of  the  transformer  is 
IP  I  Xp  H — -sj  and  is  at  right  angles  to  Ip  or  Is.  The 
impressed  primary  E.M.F.,  Ep,  is  equal  to  the  vectorial 

(r>     \  /  y-  \ 

Rp  +  -j-J,  and  Ip  I  Xp  +  -~1  as  shown  in 

Fig.  122.     Hence  the  regulation  is  expressed  by 
E    -£- 

p 
-P,          ,     , .  T 

Regulation  = 


In  practice  it  will  be  found  impossible  to  complete  the 
solution   of    these    diagrams    graphically   because   of   the 


ALTERNATING-CURRENT    MACHINES. 


extreme  flatness  of  the  triangles.     The  better  way  is  to 
draw  an  exaggerated  but  clear  diagram,  and  obtain  the 

true  values  of  the  sides  by 
the  methods  of  trigonom- 
etry and  geometry. 

The  regulation  of  a  trans- 
former at  any  load  and 
power  factor  can  be  com- 
puted when  the  equivalent 
resistance  and  the  equiva- 
lent reactance  are  known. 
The  equivalent  resistance 
can  be  determined  experi- 
mentally by  measuring  the 
primary  and  secondary  re- 
sistances using  direct  cur- 
rent, and  then  reducing  the 
latter  to  the  primary  cir- 
cuit by  dividing  by  r2. 
The  equivalent  reactance 
can  be  determined  by 
short-circuiting  one  wind- 
ing and  impressing  a  suffi- 
cient E.M.F.  upon  the 
other  to  permit  full-load 
current  to  flow.  This  cur- 
rent value  multiplied  by 
the  total  equivalent  resistance  gives  the  resistance  drop 
which  must  be  subtracted  from  the  impressed  E.M.F.  at 
the  proper  phase  angle  to  obtain  the  total  equivalent  reac- 
tance drop.  Dividing  this  by  the  current  value  there 
obtains  the  total  equivalent  reactance.  The  regulation  of 


Fig.  122. 


THE   TRANSFORMER. 


179 


the  transformer  at  any  load  and  power  factor  may  there- 
after be  calculated. 

This  method  is  more  reliable  than  the  load  test,  in  which 
the  no-load  and  full-load  voltages  are  directly  measured, 
because  of  the  magnitudes  of  these  quantities.  A  slight 
error  in  these  measurements  would  introduce  a  considerable 
error  in  the  regulation  value,  for  taking  the  difference 
between  these  large  quantities  exaggerates  the  error  of 
measurement. 

68.  Circle  Diagram.  —  The  magnitude  and  phase  of 
the  current  produced  by  a  con- 
stant impressed  primary  elec- 
tromotive force,  Ep  in  Fig.  121, 
depends  upon  the  resistance 
and  reactance  of  the  circuit. 
If  the  load  be  non-inductive, 
the  current  supplied  to  it  is  de- 
pendent upon  the  resistance  of 
the  load.  Neglecting  the  effect 
of  shunt  exciting  circuit,  the 
impressed  E.M.F.  has  two 
Fig-  ™3'  components,  that  necessary  to 

overcome  the  reactive  drop  due  to  the  leakage  flux  in 
the  transformer  itself,  and  that  necessary  to  overcome 
the  resistance  drop  due  to  the  resistance  of  the  entire 
circuit.  These  are  at  right  angles  to  each  other  and 

may    be    represented   respectively   by  Ip  ( Xp  +  -~j  and 

7?  \ 

- — -}  as  in  Fig.  123.    If  the  resistance  of  the 

load  be  altered,  the  current  will  change  and  the  point  A 


i8o 


ALTERNATING-CURRENT   MACHINES. 


will  be  in  a  different  position,  since  Xp  +  —£  is  constant. 

However,  the  impressed  E.M.F.  is  always  equal  and 
opposite  to  the  resultant  of  the  reactance  drop  and  the 
resistance  drop,  and  to  satisfy  this  condition  the  locus  of 
the  point  A  must  be  a  semicircle.  As  Ip  is  proportional 
to  the  reactive  drop,  and  since  the  two  angles  marked  0  are 
equal,  it  follows  that  the  locus  of  the  point  B  is  also  a  semi- 
circle. The  diameter  of  this  semicircle  is 


amperes, 


which  is  the  condition  corresponding  to  zero  resistance. 
To  sum  up,  then,  the  locus  of  the  load  current  for  various 
resistances,  when  the  load  is  non-inductive,  is  a  semicircle 
whose  diameter  is  the  ratio  of  the 
primary  impressed  E.M.F.  to  the 
total  equivalent  reactance  of  the 
transformer,  and  whose  diameter 
is  at  right  angles  to  Ep. 

The  total  current  produced  by 
Ep  of  Fig.  121,  when  the  load  is 
non-inductive  (X2=o),  is  the 
vectorial  sum  of  Ip  and  Iexc,  as 
shown  in  Fig.  124.  The  resulting 
primary  current  lags  behind  Ep  by 
an  angle  <j>p,  and  the  power  factor 
of  the  complete  circuit  is  the  ratio  Fig  I24 

of  OM  to  ON,  or  cos  <j>p.     The 

power  supplied  to  the  transformer  is   the  product  of  Ep 
and  OM.     Knowing  the  copper  and  core  losses,  the  out- 


THE   TRANSFORMER.  l8l 

put  P  may  be  computed,  and  the  efficiency  of  the  trans- 
former determined.     The  regulation  is  then  obtained  from 


Regulation 


-P 


P 

Is 


69,   Methods  of  Connecting  Transformers.  —  There  are 
numerous  methods  of  connecting  transformers  to  distribut- 
ing circuits.     The  simplest  case  is  that 
of    a  single    transformer  in  a  single- 
phase  circuit.      Fig.   125  shows  such 
an  arrangement.     This  and  the  suc- 
ceeding figures  have  the  pressure  and 
current  values  of  the  different  parts 
marked   on  them,  assuming  in  each 
Fis-  "5-  case  a  I-K.W.,  i  to  10  step-down  trans- 

former. As  in  Fig.  126,  two  or  more  transformers  may 
have  their  primaries  in  parallel 
on  the  same  circuit,  and  have 
their  secondaries  independent.  If 
the  two  secondaries  of  this  case 
are  connected  properly  in  series 
a  secondary  system  of  double  the 
potential  will  result,  or  by  adding 
a  third  wire  to  the  point  of  junc- 
ture, as  shown  by  the  dotted  line 
of  Fig.  127,  a  three- wire  system 
of  distribution  can  be  secured.  Fig>  "6- 

The  secondaries  must  be  connected  cumulatively;  that  is, 
their  instantaneous  E.M.F.'s  must  be  in  the  same  direction. 
If  connected  differentially,  there  would  be  no  pressure 


182 


ALTERNATING-CURRENT    MACHINES. 


Fig.   127. 


between  the  two  outside  secondary  wires,  the  instantane- 
ous pressures  of  the  two  coils  being  equal  and  opposed 
throughout  the  cycle.  Again, 
with  the  same  condition  of  pri- 
maries, the  secondaries  can  be 
connected  in  multiple  as  in  Fig. 
128.  Here  the  connections  must 
be  such  that  at  any  instant  the 
E.M.F.'s  of  the  secondaries  are 
toward  the  same  distributing  wire. 
The  connection  of  more  than  two 
secondaries  in  series  is  not  com- 
mon, but  where  a  complex  net- 
work of  secondary  distributing  mains  is  fed  at  various 
points  from  a  high-tension  system,  secondaries  are  neces- 
sarily put  in  multiple. 

In  many  types  of  modern  transformers  it  is  usual  to 
wind  the  secondaries  (low-ten- 
sion) in  two  separate  and  simi- 
lar coils,  all  four  ends  being 
brought  outside  of  the  case. 
This  allows  of  connections  to 
two-wire  systems  of  either  of 
two  pressures,  or  for  a  three- 
wire  system  according  to  Figs. 
127  and  128,  to  be  made  with 
the  one  transformer,  this  being 
more  economical  than  using 
two  transformers  of  half  the  size,  both  in  first  cost  and 
in  cost  of  operation.  In  many  transformers  the  primary 
coils  are  also  wound  in  two  parts.  In  these,  however,  the 
four  terminals  are  not  always  brought  outside,  but  in  some 


Fig.  128. 


THE   TRANSFORMER. 


183 


cases  are  led  to  a  porcelain  block  on  which  are  four  screw- 
connectors  and  a  pair  of  brass  links,  allowing  the  coils  to 
be  arranged  in  series  or  in  multiple  according  to  the  pressure 
of  the  line  to  which  they  are  to  be  connected.  From  this 
block  two  wires  run  through  suitably  bushed  holes  outside 
the  case. 

A  two-phase  four-wire  system  can  be  considered  as  two 
independent    single-phase    systems,    transformation    being 
accomplished  by  putting  similar 
single-phase  transformers  in  the 
circuit,  one  on  each  phase.     If 
it  is  desired  to  tap  a  two-phase 
circuit    to   supply   a   two-phase 
three-wire   circuit,    the  arrange- 
ment of-  Fig.    129  is  employed. 
Fig-  «g.  By     the     reverse     connections 

two-phase  three-wire  can  be  transformed  to  two-phase 
four-wire.  An  interesting  transformer  connection  is  that 
devised  by  Scott,  which  permits  of  transformation  from 
two-phase  four-wire  to  three-phase  three-wire.  Fig.  130 
shows  the  connections  of  the  two  transformers.  If  one 


Fig.  130. 


Fig.  131. 


of    the    transformers   has    a   ratio    of    10    to    i    with    a 
tap  at  the  middle  point  of  its  secondary  coil,  the  other 

—  & 


must  have  a  ratio  of  10  to 


.867  I 
\ 


10  to 


One  ter- 


1 84 


ALTERNATING-CURRENT   MACHINES. 


minal  of  the  secondary  of  the  latter  is  connected  to  the 
middle  of  the  former,  the  remaining  three  free  terminals 
being  connected  respectively  to  the  three-phase  wires.  In 
Fig.  131,  considering  the  secondary  coils  only,  let  mn  rep- 
resent the  pressure  generated  in  the  first  transformer. 
The  pressure  in  the  second  transformer  is  at  right  angles 
(§7)  to  that  in  the  first,  and  because  of  the  manner  of 
connection,  proceeds  from  the  center  of  mn.  Therefore 
the  line  op  represents  in  position,  direction,  and  magnitude 
the  pressure  generated  in  the  second.  From  the  geo- 
metric conditions  mnp  is  an  equilateral  triangle,  and  the 
pressures  represented  by  the  three  sides  are  equal  and  at 
60°  with  the  others.  This  is  suitable  for  supplying  a 
three-phase  system.  In  power  transmission  plants  it  is 
not  uncommon  to  find  the  generators  wound  two-phase, 
and  the  step-up  transformers  arranged  to  feed  a  three- 
phase  line. 

In  America  it  is  common  to  use  one  transformer  for 
each  phase  of  a  three-phase  circuit.  The  three  transform- 
ers may  be  connected  either 
Y  or  A.  They  may  be  Y  on 
the  primary  and  A  on  the 
secondary,  or  vice  versa. 
Fig.  132  shows  both  primary 
and  secondary  connected  A. 
The  pressure  on  each  pri- 
mary is  1000  volts,  and  as 
a  I-K.W.  transformer  was 
assumed,  i.e.,  i  K.W.  per  phase,  there  will  be  one  ampere 
in  each,  calling  for  1.7  0/3)  amperes  in  each  primary 
main  (§  45).  This  arrangement  is  most  desirable  where 
continuity  of  service  is  requisite,  for  one  of  the  trans- 


Fig,  132. 


THE   TRANSFORMER. 


I8S 


Fig.  133- 


formers  may  be  cut  out  and  the  system  still  be  opera- 
tive, the  remaining  transformers  each  taking  up  the  dif- 
ference between  J  and  \  the  full  load ;  that  is,  if  the 
system  was  running  at  full  load,  and  one  transformer  was 

cut  out,  the  other  two 
would  be  overloaded  i6f 
Per  cent.  Even  if  two  of 
^~OAt  them  were  cut  out,  ser- 
vice over  the  remaining 
phase  could  be  main- 
tained. It  is  not  uncom- 
mon to  regularly  supply 
motors  from  three-phase 
mains  by  two  somewhat 
larger  transformers  rather  than  by  three  smaller  ones. 
Fig.  133  shows  the  connections  for  both  primaries  and 
secondaries  in  Y.  If  in  this  arrangement  one  transformer 
be  cut  out,  one  wire  of  the  system  becomes  idle,  and  only 
a  reduced  pressure  can  be  maintained  on  the  remaining 
phase.  The  advantage  of  the  star  connection  lies  in  the 
fact  that  each  transformer  need  be  wound  for  only  57.7 
per  cent  of  the  line  voltage. 
In  high-tension  transmis- 
sion this  admits  of  build- 
ing the  transformers  much 
smaller  than  would  be 
necessary  if  they  were  A 
connected.  Fig.  134  shows 
the  connections  for  prima- 
ries in  A,  secondaries  in  Fig-  I34- 
Y;  and  Fig.  135  those  for  primaries  in  Y  and  secondaries 
in  A.  By  taking  advantage  of  these  last  two  arrange- 


186          ALTERNATING-CURRENT   MACHINES. 


ments,  it  is  possible  to  raise  or  lower  the  voltage  with  i 

to    i    transformers.      With    three    i    to    i    transformers, 

arranged  as  in  Fig.  134,   100 

volts   can    be    transformed    to 

173  volts;  while  if  connected  as 

in  Fig.  135,  100  volts  can  be 

transformed  down  to  58  volts. 

Fig.  136  shows  a  transformer 
and  another  one  connected  as 
an  autotransformer  doing  the 
same  work.  Since  the  required 
ratio  of  transformation  is  i  to 
2,  the  autotransformer  does  the  work  of  the  regular  trans- 
former with  one-half  the  first  cost,  one-half  the  losses, 
and  one-half  the  drop  in  potential  (regulation).  The  only 
objection  to  this  method  of  transformation  is  that  the  pri- 
mary and  secondary  circuits  are  not  separate.  With  the 
circuits  grounded  at  certain  points,  there  is  danger  that  the 
insulation  of  the  low-tension  circuit  may  be  subjected  to 


Fig.   135- 


Losses  not  cansideired 


Losses  not  considered. 
Fig.  136. 


the  voltage  of  the  high-tension  circuit.  One  coil  of  an 
autotransformer  must  be  wound  for  the  lower  voltage,  and 
the  other  coil  for  the  difference  between  the  two  voltages 
of  transformation.  The  capacity  of  an  autotransformer  is 
found  by  multiplying  the  high-tension  current  by  the  dif- 
ference between  the  two  operative  voltages.  Autotransfor- 
mers  are  often  called  compensators.  Compensators  are 


THE  TRANSFORMER. 


I87 


advantageously  used  where  it  is  desired  to  raise  the  poten- 
tial by  a  small  amount,  as  in  boosting  pressure  for  very 
long  feeders.  Fig.  137  shows  three  i  to  2  transformers 


Three  16.5  Kv/. 
Transformers 
Ratio  1  to  1 


Loss.es  .(rot  considered 


Losses  not  considered 
Fig.  137- 


connected  in  A  on  a  three-phase  system,  and  three  i  to  i 
compensators  connected  in  Y  to  do  the  same  work. 

From  a  two-phase  circuit,  a  single-phase  E.M.F.  of  any 
desired  magnitude  and  any  desired  phase-angle  may  be 
secured  by  means  of  suitable  transformers,  as  shown  in 
Fig.  138.  Suppose  the  two  phases  X  and  Y  of  a  two-phase 
system  be  of  100  volts  pressure,  and  it  is  desired  to  obtain 
a  single-phase  E.M.F.  of  1000  volts  and  leading  the  phase 
-X"  by  30°.  As  in  Fig.  139,  draw  a  line  representing  the 


DIRECTION  OF  PHASE  X. 

Fig.  139. 


direction  of  phase  X.  At  right  angles  thereto,  draw  a  line 
representing  the  direction  of  phase  Y.  From  their  inter- 
section draw  a  line  1000  units  long,  making  an  angle  of 


188          ALTERNATING-CURRENT   MACHINES. 

30°  with  X.  It  represents  in  direction  and  in  length  the 
phase  and  the  pressure  of  the  required  E.M.F.  Resolve 
this  line  into  components  along  X  and  F,  and  it  becomes 
evident  that  the  secondary  of  the  transformer  connected 
to  X  must  supply  the  secondary  circuit  with  866  volts, 
and  that  the  secondary  of  the  other  must  supply  500  volts. 
Therefore  the  transformer  connected  to  X  must  step-up 
i  to  8.66  and  that  connected  to  F  must  step-up  i  to  5.  If 
10  amperes  be  the  full  load  on  the  secondary  circuit,  the 
first  transformer  must  have  a  capacity  of  8.66  K.W.,  and  the 
second  a  capacity  of  5  K.W.  The  load  on  X  and  F  is  not 
balanced. 

70.  Lighting  Transformers.  —  Because  of  the  extensive 
use  of  transformers  on  distributing  systems  for  electric 
lighting,  the  various  manufacturers  have  to  a  great  extent 
standardized  their  lines  of  lighting  transformers.  Some  of 
these  will  be  briefly  described. 

The  Wagner  Electric  Mfg.  Co.'s  "type  M"  transformer 
is  illustrated  in  Fig.  140.  It  is  of  the  shell  type  of  con- 
struction, makers  of  this  type  claiming  for  it  superiority 
of  regulation  and  cool  running.  In  the  shell  type  the  iron 
is  cooler  than  the  rest  of  the  transformer,  in  the  core  type 
it  is  hotter.  As  the  "ageing"  of  the  iron,  or  the  increase 
of  hysteretic  coefficient  with  time,  is  believed  to  be  aggra- 
vated by  heat,  this  is  claimed  as  a  point  of  superiority  of 
the  shell  type.  However,  the  prime  object  in  keeping  a 
transformer  cool  is  not  to  save  the  iron,  but  to  protect  the 
insulation;  and  as  the  core  type  has  less  iron  and  generally 
less  iron  loss,  the  advantages  do  not  seem  to  be  remarkably 
in  favor  of  either.  In  the  Wagner  "type  M"  transformers 
the  usual  practice  of  having  two  sets  of  primaries  and  sec- 


THE   TRANSFORMER, 


189 


Tig,  140. 


Fig.  141. 


IQO  ALTERNATING-CURRENT   MACHINES. 

ondaries  is  followed.  Fig.  141  shows  the  three  coils  com- 
posing one  set.  A  low-tension  coil  is  situated  between  two 
high-tension  coils,  this  arrangement  being  conducive  to  a 
good  regulation.  The  ideal  method  would  be  to  have  the 
coils  still  more  subdivided  and  interspersed,  but  practical 
reasons  prohibit  this.  The  space  between  the  coils  and 
the  iron  is  left  to  facilitate  the  circulation  of  the  oil  in  which 
they  are  submerged.  The  laminae  for  the  shell  are  stamped 
each  in  two  parts  and  assembled  with  joints  staggered. 
As  can  be  seen  from  the  first  cut,  all  the  terminals  of  the 
two  primary  and  the  two  secondary  coils  are  brought  outside 
the  case.  The  smaller  sizes  of  this  line  of  transformers, 
those  under  1.5  K.W.,  have  sufficient  area  to  allow  their 
running  without  oil,  so  the  manufacturers  are  enabled  to 
fill  the  retaining  case  with  an  insulating  compound  which 
hardens  on  cooling. 

The  General  Electric  Co.'s  "H"  transformers  are  of 
the  core  type.  In  Fig.  142 
is  shown  a  sectional  view 
giving  a  good  idea  of  the 
arrangement  of  parts  in 
this  type.  Fig.  103  is  also 
one  of  this  line  of  trans- 
formers. In  it  is  shown 
the  tablet  board  of  porce- 
lain on  which  the  connec- 
tions of  the  two  high-ten- 
sion coils  may  be  changed 
from  series  to  parallel  or  Flg>  I42' 

vice  versa,  so  that  only  two  high-tension  wires  are  brought 
through  the  case.  Fig.  143  shows  the  arrangement  of  the 
various  parts  in  the  assembled  apparatus.  The  makers 


THE   TRANSFORMER. 


191 


claim  for  this  type  that  the  coils  run  cooler  because  of 
their  being  more  thoroughly  surrounded  with  oil  than 
those  of  the  shell  type.  Another  .point  brought  forward 
is  that  copper  is  a  better  conductor  of  heat  than  iron; 'the 
heat  from  the  inner  portions 
of  the  apparatus  is  more  readily 
dissipated  than  in  the  shell 
type.  The  core  has  the  advan- 
tage of  being  made  up  of  simple 
rectangular  punchings,  and  the 
disadvantage  of  having  four 
instead  of  two  joints  in  the 
magnetic  circuit.  A  particular 
advantage  of  the  "type  H" 
transformer  is  the  ease  and 
certainty  with  which  the  pri- 
Flg'  I43'  mary  windings  can  be  sepa- 

rated from  the  secondary  windings.  A  properly  formed 
seamless  cylinder  of  fiber  can  be  slipped  over  the  inner 
winding  and  the  outer  one  wound  over  it.  This  is  much 
more  secure  than  tape  or  other  material  that  has  to  be 
wound  on  the  coils. 

Fig.  144  shows  a  2-K.w.  O.  D.  transformer  without  the 
case.  A  tablet  board  is  used  for  the  terminals  of  the  high- 
tension  coils,  but  the  low-tension  wires  are  all  run  out  of  the 
case.  Fig.  145  shows  one  of  the  coils.  "Type  O.  D."  trans- 
formers are  built  from  J  to  25  K.W.  for  lighting  and  to  50 
K.W.  for  power.  Those  of  10  K.W.  or  less  are  in  cast-iron 
cases,  those  above  10  K.W.  in  corrugated  iron  cases  with 
cast  tops  and  bottoms.  The  corrugations  quite  materially 
increase  the  radiating  surface.  The  windings  are  sub- 
merged in  oil. 


IQ2          ALTERNATING-CURRENT   MACHINES. 

An  example  of  the  Stanley  Electric  Manufacturing  Co.'s 
standard  line  of  "type  A.   O."  transformers  is  given  in 


Fig.  144. 

Fig.  146.     These  are  also  of  the  shell  type,  with  divided 
primaries  and  secondaries. 

71.  Cooling  of  Transformers.  —  The  use  of  oil  to  assist 
in  the  dissipation  of  the  heat  produced  during  the  opera- 
tions of  transformers  is  almost  universal  in  sizes  of  less 
than  about  100  K.W.,  especially  if  designed  for  outdoor 
use.  Some  small  transformers  are  designed  to  be  self- 
ventilating,  taking  air  in  at  the  bottom,  which  goes  out  at 
top  as  a  result  of  being  heated.  They  are  not  well  pro- 
tected from  the  weather,  and  are  liable  to  have  the  natural 
draft  cut  off  by  the  building  of  insects'  nests.  Larger 


THE   TRANSFORMER. 


193 


Fig.   145. 


transformers  that  are  air  cooled  and  that  supply  their  own 
draft  are  used  to  some  extent  in  central  stations  and  other 


Fig.  146. 

places  where  they  can  be  properly  protected  and  attended 
to.  A  forced  draft  is,  however,  the  more  common.  Where 
such  transformers  are  employed,  there  are  usually  a  number 


194 


ALTERNATING-CURRENT  MACHINES. 


of  them  ;  and  they  are  all  set  up  over  a  large  chamber  into 
which  air  is  forced  by  a  blower,  as  indicated  in  Fig.  147. 


Fig.  147. 


Dampers  regulate  the  flow  of  air  through  the  transformers. 
They  can  be  adjusted  so  that  each  transformer  gets  its 
proper  share. 

Fig.  148  shows  a  General  Electric  Company's  air-blast 
transformer  in  process  of  construction.  The  iron  core  is 
built  up  with  spaces  between  the  laminae  at  intervals  ;  and 
the  coils,  which  are  wound  very  thin,  are  assembled  in 
small  intermixed  groups  with  air  spaces  maintained  by 
pieces  of  insulation  between  them.  The  assembled  struc- 
ture is  subjected  to  heavy  pressure,  and  is  bound  together 
to  prevent  the  possibility  of  vibration  in  the  coils  due  to 
the  periodic  tendency  to  repulsion  between  the  primary  and 
the  secondary.  These  transformers  are  made  in  sizes  from 
100  K.W.  to  1000  K.W.  and  for  pressures  up  to  35,000  volts. 

Another  method  of  cooling  a  large  oil  transformer  is  to 
circulate  the  oil  by  means  of  a  pump,  passing  it  through  a 
radiator  where  it  can  dissipate  its  heat.  Again  cold  water 
is  forced  through  coils  of  pipe  in  the  transformer  case,  and 
it  takes  up  the  heat  from  the  oil.  There  is  the  slight  dan- 
ger in  this  method  that  the  pipes  may  leak  and  the  water 
may  injure  the  insulation.  Water-cooled  transformers 
have  been  built  up  to  2000  K.W.  capacity. 


THE   TRANSFORMER.  195 

In  those  cases  where  the  transformer  requires  some 
outside  power  for  the  operation  of  a  blower  or  a  pump, 
the  power  thus  used  must  be  charged  against  the  trans- 


Fig.  148. 

former  when  calculating  its  efficiency.  In  general  this 
power  will  be  considerably  less  than  I  %  of  the  trans- 
former capacity. 

72.  Constant-Current  Transformers.  —  For  operating 
series  arc-light  circuits  from  constant  potential  alternating- 
current  mains,  a  device  called  a  constant-current  trans- 
former is  frequently  employed.  A  sketch  showing  the 
principle  of  operation  is  given  in  Fig.  149.  A  primary 
coil  is  fixed  relative  to  the  core,  while  a  secondary  coil  is 


196 


ALTERNATING-CURRENT  MACHINES. 


allowed  room  to  move  from  a  close  contact  with  the 
primary  to  a  considerable  distance  from  it.  This  secon- 
dary coil  is  nearly  but  not  entirely 
counter-balanced.  If  no  current 
is  taken  off  the  secondary  that 
coil  rests  upon  the  primary. 
When,  however,  a  current  flows 
in  the  two  coils  there  is  a  repul- 
sion between  them.  The  counter- 
poise is  so  adjusted  that  there  is 
an  equilibrium  when  the  current 

is  at  the  proper  value.  If  the  current  rises  above  this 
value  the  coil  moves  farther  away,  and  there  is  an  increased 
amount  of  leakage  flux.  This  lowers  the  E.M.F.  induced 


Fig.  150 


THE   TRANSFORMER. 


197 


in  the  secondary,  and  the  current  falls  to  its  normal  value. 
Thus  the  transformer  automatically  delivers  a  constant 
current  from  its  secondary  when  a  constant  potential  is 
impressed  on  its  primary. 

Fig.  150  shows  the  mechanism  of  such  an  apparatus  as 
made  by  the  General  Electric  Company.  The  cut  is  self- 
explanatory.  Care  is  taken  to  have  the  leads  to  the 


ing  coil  very  flexible.  Transformers  for  50  lamps  or 
more  are  made  with  two  sets  of  coils,  one  primary  coil 
being  at  the  bottom,  the  other  at  the  top.  The  moving 
coils  are  balanced  one  against  the  other,  avoiding  the 
necessity  of  a  very  heavy  counterweight.  Fig.  151  shows 
a  5o-light  constant-current  transformer  without  its  case. 
Fig.  152  shows  a  complete  2 5 -lamp  apparatus.  The  tank 


198  ALTERNATING-CURRENT   MACHINES. 

is  filled  with  oil,  the  same  as  an  ordinary  transformer. 
Great  care  must  be  taken  to  keep  these  transformers  level, 
and  to  assist  in  this  the  larger  sizes  have  spirit-levels  built 


Fig.  152. 

into  the  case.  A  pair  of  these  transformers  can  be  spe- 
cially wound  and  connected  to  supply  a  series  arc-light 
circuit  from  a  three-phase  line,  keeping  a  balanced  load  on 
the  latter. 

73.  Polyphase  Transformers.  —  In  transforming  from 
one  w-phase  system  to  another  w-phase  system,  instead  of 
using  n  single-phase  transformers,  one  w-phase  trans- 
former may  be  employed.  A  polyphase  transformer  con- 
sists of  several  single-phase  transformers  having  portions  of 
their  magnetic  circuits  in  common.  As  these  common 
portions  of  the  magnetic  circuits  carry  fluxes  differing  in 


THE  TRANSFORMER. 


I99 


phase,  an  economy  of  material  results  due  to  the  fact  that 
the  resultant  flux  is  less  than  the  arithmetical  sum  of  the 
component  fluxes.  A  further  saving  is  effected  due  to  the 


Fig.  153- 

necessity  of  only  one  instead  of  several  containing  cases, 
but  this  disappears,  however,  when  the  single-phase  trans- 
formers are  all  mounted  in  one  case. 

Three-phase  transformers  are  used  extensively  in  Europe 


Fig-   154- 


and  the  tendency  toward  their  use  in  America  is  constantly 
increasing.  The  magnetic  circuits  of  the  two  most  common 
types  of  three-phase  transformers  are  diagrammatically 


200  ALTERNATING-CURRENT   MACHINES. 

shown  in  Fig.  153  and  Fig.  154.     The  first  is  known  as  the 
core  type,  and  the  other  as  the  shell  type. 

The  advantages  claimed  for  the  three-phase  transformer 
over  three  single-phase  units  are:  i,  a  saving  of  about  15  % 
in  first  cost;  2,  the  required  floor  space  is  smaller  and  the 
weight  is  less;  3,  greater  efficiency.  In  the  case  of  a  break- 
down, however,  the  resulting  derangement  of  the  service 
and  the  cost  of  repair  are  greater  for  three-phase  than  for 
single-phase  transformers.  Another  disadvantage  is  the 
greater  cost  of  a  spare  unit.  In  large  power  stations  an 
installation  of  three-phase  transformers  is  believed  to  be 
more  economical  than  an  installation  of  single  phase  units. 

PROBLEMS. 

1.  Determine  the  total  flux  of  a  60  cycle  lighting  transformer  having 
0.8  primary  turn  per  volt  of  primary  impressed  E.M.F. 

2.  Calculate  the  eddy  current  and  hysteresis  losses  in  the  iron  of  a 
125  cycle  core-type  transformer  for  which  <£m  is  0.2  megamaxwell.    The 
mean  length  of  the  magnetic  circuit  is  35  inches  and  the  cross-section  is 
9  square  inches. 

3.  If  the  E.M.F.  impressed  upon  the  primary  of  the  transformer  of 
the  preceding  problem  be  2200  volts,  compute  the  value  of  the  exciting 
current  and  its  phase. 

4.  A  transformer  has  2000  turns  of  No.  16  B.  &  S.  copper  wire  on  the 
secondary  winding,  and  100  turns  of  No.  4  B.  &  S.  copper  wire  on  the 
primary  winding.     The  mean  lengths  of  the  secondary  and  primary 
turns  are  respectively  17  and  28  inches.     Determine  the  total  equiva- 
lent primary  resistance. 

5.  What  is  the  copper  loss  in  the  transformer  of  the  preceding  problem 
when  the  primary  current  is  25  amperes,  the  exciting  current  being 
neglected  ? 

6.  Assuming  that  the  transformer  of  problems  2  and  3  is  a  5  K.W., 
20  :  i   step-down  transformer,  and  that  the  primary  and  secondary 
resistances  are  16.6  and  .041  ohms  respectively;  determine  the  efficiency 
at  full  load. 


PROBLEMS.  201 

7.  Find  the  all-day  efficiency  of  the  transformer  of  the  preceding 
problem,  basing  the  calculation  upon  5  hours  full -load  and  19  hours 
no-load  operation. 

8.  Calculate  the  equivalent  primary  and  the  equivalent  secondary 
leakage  reactances  of  a  60  ^  shell  type  transformer  having  one  primary 
and  one  secondary  coil.     The  constants  indicated  in  Fig.  117  are: 

Mean  length  of  secondary  turn  28  inches 
Mean  length  of  primary  turn  37.5  inches 
np  —  396  P  =  S  =  1.25  inches 

ns   =    18  g  =      .25  inch 

A    =    15.5  inches  /  =    6.5    inches. 

What  is  the  total  equivalent  primary  leakage  reactance,  the  transformer 
assumed  to  be  under  considerable  load  ? 

9.  The  resistances  of  the  primary  and  secondary  windings  of  a  i  :  2, 
60  cycle,  step-up  transformer  are  respectively  o.i  and  .34  ohms.     The 
equivalent  leakage  reactances  of  the  primary  and  of  the  secondary  are 
.14  and  .5  ohms  respectively,  and  the  secondary  induced  electromotive 
force,  Es,  is  220  volts.     Determine  the  E.M.F.  to  be  impressed  upon 
the  primary  terminals,  the  load  on  the  secondary  consisting  of  a  6  ohm 
resistance  and  an  inductance  of  .01  henry.     The  exciting  current  is  .85 
amperes  and  lags  70°  behind  EQ. 

10.  Calculate  the  regulation  of  the  transformer  of  the  preceding 
problem. 

n.  It  is  desired  to  transform  from  2200  volts  two-phase  to  500  volts 
three-phase  by  means  of  a  Scott  transformer.  Allowing  one  volt  per 
turn  on  the  windings,  find  the  number  of  turns  on  each  primary  and  on 
each  secondary  coil. 

12.  A  loo-iooo  volt  step-up  transformer  is  connected  to  the  A  -phase 
of  a  two-phase,  four-wire  system,  and  a  100-2000  volt  step-up  trans- 
former is  connected  to  the  other  phase.  Determine  the  magnitude  and 
phase  of  the  secondary  electromotive  force  when  the  secondary  coils 
are  in  series. 


202  ALTERNATING-CURRENT   MACHINES. 


CHAPTER    VII. 

MOTORS. 
INDUCTION   MOTORS. 

74.  Rotating  Field.  —  Suppose  an  iron  frame,  as  in  Fig. 
155,  to  be  provided  with  inwardly  projecting  poles,  and  that 
these  be  divided  into  three  groups,  arranged  as  in  the  dia- 
gram, poles  of  the  same  group 
being  marked  by  the  same 
letter.  If  the  poles  of  each 
group  be  alternately  wound 
in  opposite  directions,  and  be 
connected  to  a  single  source 
of  E.M.F.,  then  the  resulting  current 
would  magnetize  the  interior  faces  al- 
ternately north  and  south.  If  the  im- 
pressed E.M.F.  were  alternating,  then 
the  polarity  of  each  pole  would  change  Flg>  I55' 

with  each  half  cycle.  If  the  three  groups  of  windings 
be  connected  respectively  with  the  three  terminals  of  a 
three-phase  supply  circuit,  any  three  successive  poles  will 
assume  successively  a  maximum  polarity  of  the  same 
sign,  the  interval  required  to  pass  from  one  pole  to  its 
neighbor  being  one-third  of  the  duration  of  a  half  cycle. 
The  maximum  intensity  of  either  polarity  is  therefore 
passed  from  one  pole  to  the  next,  and  the  result  is  a  rotat- 
ing field.  If  the  frequency  of  the  supply  E.M.F.  be/,  and 


MOTORS.  203 

if  there  be  p  pairs  of  poles  per  phase,  then  the  field  will 
make  one  complete  revolution  in  j  seconds.    It  will  there- 
fore make  —  =  —  complete    revolutions    per   second.     A 
p       oo 

rotating  field  can  be  obtained  from  any  polyphase  supply- 
circuit  by  making  use  of  appropriate  windings. 


75.  The  Induction  Motor.  —  If  a  suitably  mounted 
hollow  conducting  cylinder  be  placed  inside  a  rotating  field, 
it  will  have  currents  induced  in  it,  due  to  the  relative 
motion  between  it  and  the  field  whose  flux  cuts  the  surface 
of  the  cylinder.  The  currents  in  combination  with  the  flux 
will  react,  and  produce  a  rotation  of  the  cylinder.  As  the 
current  is  not  restrained  as  to  the  direction  of  its  path,  all 
of  the  force  exerted  between  it  and  the  field  will  not  be  in 
a  tangential  direction  so  as  to  be  useful  in  producing  rota- 
tion. This  difficulty  can  be  overcome  by  slotting  the 
cylinder  in  a  direction  parallel  with  the  axis  of  revolution. 
Nor  will  the  torque  exerted  be  as  great  as  it  would  be  if 
the  cylinder  were  mounted  upon  a  laminated  iron  core. 
Such  a  core  would  furnish  a  path  of  low  reluctance  for  the 
flux  between  poles  of  opposite  sign.  The  flux  for  a  given 
magnetomotive  force  would  thereby  be  greater,  and  the 
torque  would  be  increased. 

Induction  motors  operate  according  to  these  principles. 
The  stationary  part  of  an  induction  motor  is  called  the 
stator,  and  the  moving  part  is  called  the  rotor.  It  is  com- 
mon practice  to  produce  the  rotating  field  by  impressing 
E.M.F.  upon  the  windings  of  the  stator.  There  are, 
however,  motors  whose  rotating  fields  are  produced  by  the 
currents  in  the  rotor  windings. 


204 


ALTERNATING-CURRENT    MACHINES. 


The  construction  of  a  line  of  induction  motors  manu- 
factured by  the  General  Electric  Company  is  shown  in 
Fig.  156.  In  this  type  the  outer  edges  of  the  stator  lami- 
nations are  directly  exposed  to  the  air,  thus  improving 
ventilation.  The  stator  core  and  windings  of  a  Westing- 


Fig.  156. 

house  induction  motor  are  shown  in  Fig.  157.  Each  pro- 
jection of  the  core  does  not  necessarily  mean  a  pole;  for 
it  is  customary  to  employ  a  distributed  winding,  there 
being  several  slots  per  pole  per  phase.  The  stator  wind- 
ings are  similar  to  the  armature  windings  of  polyphase 
alternators.  The  winding  for  each  phase  consists  of  a 


MOTORS. 


205 


group  of  coils,  one  group  for  each  pole.     The  individual 
coils  of  each  group  are  laid  in  separate  slots.     The  stator 
windings  of  a  three-phase  induction  motor  are  shown  in 
Fig.     158,    where    each    loop 
represents   the   group   of  coils 
for  one  pole. 

One  type  of  rotor  is  shown 
in  Fig.  159.  The  inductors 
are  copper  bars  embedded  in 
slots  in  the  laminated  steel 
core.  They  are  all  connected, 
in  parallel,  to  copper  collars  or 
short-circuiting  rings,  one  at 
each  end  of  the  rotor.  They 
offer  but  a  very  small  resist- 
ance, and  the  currents  induced 

in  them  are  forced  to   flow  in   a  direction   parallel  with 
the   axis.     The    reaction    against  the  field  flux  is  there- 


Fig.  157. 


fore  in  a  proper  direction  to  be  most  efficient  in  producing 
rotation.  A  rotor  or  armature  of  this  type  is  called  a 
squirrel  cage. 


206  ALTERNATING-CURRENT   MACHINES. 

Another  type  of  rotor  frequently  used,  especially  in 
large  induction  motors,  has  polar  windings  which  are 
similar  to  the  windings  on  the  stator.  Fig.  160  shows  a 


Fig. 159. 


rotor  of  this  type  made  by  the  General  Electric  Company. 
The  windings  are  short-circuited  through  an  adjustable 
resistance  carried  on  the  rotor  spider.  When  starting  the 


Fig.  160. 

motor,  all  the  resistance  is  in  circuit,  and  after  the  proper 
speed  has  been  attained,  the  resistance  may  be  cut  out  by 
pushing  a  knob  on  the  end  of  the  shaft,  as  shown  in  the 
figure.  This  arrangement  permits  of  a  small  starting 
current  under  load  and  a  large  torque,  §  77.  Fig.  161 


MOTORS.  207 

shows  another  rotor  of  this  type  made  by  the  same  com- 
pany; the  windings  are  identical  with  those  on  the  other, 
except  that  their  terminals  are  brought  out  to  three  slip- 


Fig.  161. 

rings.  A  starting  resistance  can  be  placed  away  from  the 
motor  and  be  connected  with  the  rotor  windings  by  means 
of  brushes  rubbing  upon  the  slip-rings. 

76.  Starting  of  Squirrel-Cage  Motors.  —  To  avoid  the 
excessive  rush  of  current  which  would  result  from  connec- 
tion of  a  loaded  squirrel-cage  motor  to  a  supply  circuit,  use 
is  made  by  both  the  Westinghouse  Company  and  the 
General  Electric  Company  of  starting  compensators. 
These  are  auto-transformers  which  are  connected  between 
the  supply  mains,  and  which,  through  taps,  furnish  to  the 
motor  circuits  currents  at  a  lower  voltage  than  that  of 
the  supply  mains.  After  the  rotor  has  attained  the  speed 
appropriate  to  the  higher  voltage,  the  motor  connections 
are  transferred  to  the  mains,  and  the  compensator  is 
thrown  out  of  circuit.  The  connections  are  shown  in  Fig. 
162,  and  the  appearance  of  the  General  Electric  Company 
compensator  is  shown  in  Fig.  163.  The  change  of  con- 


208 


ALTERNATING-CURRENT  MACHINES. 


nections  is  accomplished  by  moving  the  handle  shown  at 
the  right  of  the  figure.  While  the  compensator  is  supplied 
v;ith  various  taps,  only  that  one  which  is  most  suitable  for 


nerstor 

Rurmirvj 

s 

de 

1 

CH  0-J  0-J  O-1  CM 


Compen  sator  winding 

Fig.  162. 

the  work  is  used  when  once  installed.  The  Westinghouse 
starter  for  squirrel-cage  induction  motors  is  shown  in 
Fig.  164.  It  consists  of  auto-transformers  and  a  multi- 
point drum-type  switch,  the  latter  being  oil  immersed  so 
as  to  eliminate  sparking  at  the  points  of  contact.  An 
important  feature  of  this  design  is  that  the  handle  is  moved 
in  but  one  direction  in  passing  from  the  off,  through  the 
starting,  to  the  running  position,  thus  making  it  impossible 
to  connect  the  motor  directly  to  the  full  line  voltage. 

Where  special  step-down  transformers  are  used  for  indi- 
vidual motors,  or  where  several  motors  are  located  close  to 
and  operated  from  a  bank  of  transformers,  it  is  sometimes 
practical  to  bring  out  taps  from  the  secondary  winding,  and 
use  a  double-throw  motor  switch,  thereby  making  provision 


MOTORS. 


209 


Fig.  163. 


Fig.  164. 


210  ALTERNATING-CURRENT   MACHINES. 

for  starting  the  motor  at  low  voltage,  while  avoiding  the 
necessity  for  a  starting  compensator. 

The  General  Electric  Company  make  small  squirrel- 
cage  motors,  with  centrifugal  friction  clutch  pulleys;  so 
that  although  a  load  may  be  belted  to  the  motor,  it  is  not 
applied  to  the  rotor  until  the  latter  has  reached  a  certain 
speed.  The  starting  current  is  therefore  a  no-load  starting 
current. 

77.   Principle  of  Operation  of  the  Induction  Motor.  —  If 

the  speed  of  rotation  of  the  field  be  V  R.P.M.  and  that  of 
the  rotor  be  V  R.P.M.,  then  the  relative  speed  between  a 
given  inductor  on  the  rotor  and  the  rotating  field  will  be 
V-  V  R.P.M.  The  ratio  of  this  speed  to  that  of  the 

V—  V 
field,  viz.,  — — =  s,  is  termed  the  slip,  and  is  generally 

expressed  as  a  per  cent  of  the  synchronous  speed.  If  the 
flux  from  a  single  north  pole  of  the  stator  be  <I>  maxwells, 
then  the  effective  E.M.F.  induced  in  a  single  rotor  inductor 

y 
is   2.22   p$  s  —~  io~8,  where   p  represents   the   number  of 

60 

pairs  of  stator  poles.  The  frequency  of  this  induced 
E.M.F.  is  different  from  that  of  the  E.M.F.  impressed 
upon  the  stator.  It  is  s  times  the  latter  frequency.  The 
frequency  would  be  zero  if  the  rotor  revolved  in  synchro- 
nism with  the  field,  and  would  be  that  of  the  field  current 
if  the  rotor  were  stationary.  As  the  slip  of  modern 
machines  is  but  a  few  per  cent  (2  %  to  15  %),  the  frequency 
of  the  E.M.F.  in  the  rotor  inductors,  under  operative  con- 
ditions, is  quite  low.  The  current  which  will  flow  in  a 
given  inductor  of  a  squirrel-cage  rotor  is  difficult  to  deter- 
mine. All  the  inductors  have  E.M.F.'s  in  them,  which  at 


MOTORS.  211 

any  instant  are  of  different  values,  and  in  some  of  them  the 
current  may  flow  in  opposition  to  the  E.M.F.  It  can  be 
seen,  however,  that  the  rotor  impedance  is  very  small.  As 
the  impedance  is  dependent  upon  the  frequency,  it  will  be 
larger  when  the  rotor  is  at  rest  than  when  revolving.  It 
will  reduce  to  the  simple  resistance  when  the  rotor  is 
revolving  in  synchronism.  Suppose  a  rotor  to  be  running 
light  without  load.  It  will  revolve  but  slightly  slower  than 
the  revolving  field,  so  that  just  enough  E.M.F.  is  generated 
to  produce  such  a  current  in  the  rotor  inductors  that  the 
electrical  power  is  equal  to  the  losses  due  to  friction,  wind- 
age, and  the  core  and  copper  losses  of  the  rotor.  If  now  a 
mechanical  load  be  applied  to  the  pulley  of  the  rotor,  the 
speed  will  drop,  i.e.,  the  slip  will  increase.  The  E.M.F. 
and  current  in  the  rotor  will  increase  also,  and  the  rotor 
will  receive  additional  electrical  power,  equivalent  to  the 
increase  in  load.  The  induction  motor  operates  in  this 
respect  like  a  shunt  motor  on  a  constant  potential  direct- 
current  circuit.  If  the  strength  of  the  rotating  field,  which 
cuts  the  rotor  inductors,  were  maintained  constant,  the  slip, 
the  rotor  E.M.F.,  and  the  rotor  current  would  vary  directly 
as  the  mechanical  torque  exerted.  If  the  rotor  resistance 
were  increased,  the  same  torque  would  require  an  increase 
of  slip  to  produce  the  increased  E.M.F.  necessary  to  send 
the  same  current,  but  the  strict  proportionality  would  be 
maintained.  The  rotating  magnetism,  which  cuts  the  rotor 
inductors,  does  not,  however,  remain  constant  under  vary- 
ing loads.  As  the  slip  increases,  more  and  more  of  the 
stator  flux  passes  between  the  stator  and  rotor  windings, 
without  linking  them.  This  increase  of  magnetic  leakage 
is  due  to  the  cross  magnetizing  action  of  the  increased 
rotor  currents.  The  decrease  of  linked  field  flux  not  only 


212 


ALTERNATING-CURRENT  MACHINES. 


lessens  the  torque  for  the  same  rotor  current,  but  also 
makes  a  greater  slip  necessary  to  produce  the  same  cur- 
rent. The  relation  which  exists  between  torque  and  slip 
for  various  rotor  resistances  is  shown  in  Fig.  165,  where 
the  full  lines  represent  torque,  and  the  dotted  lines  current. 
An  inspection  of  the  curves  shows  that  the  maximum 


torque  which  a  motor  can  give  is  the  same  for  different 
rotor  resistances.  The  speed  of  the  rotor,  however,  when 
the  motor  is  exerting  this  maximum  torque,  is  different  for 
different  resistances.  This  fact  is  made  use  of  in  starting 
induction  motors  having  wound  rotors  so  that  the  starting 
current  may  not  be  excessive.  The  rotor  resistance  is 
designed  to  give  full-load  torque  at  starting  with  full  load 
current.  When  the  motor  reaches  its  proper  speed,  this 
resistance  is  gradually  cut  out  so  that  a  large  torque  is 
secured  within  the  operative  range. 


MOTORS.  213 

78.  Relation  between  Speed  and  Efficiency.  —  A  portion 
of  the  total  power  supplied  to  the  stator  of  an  induction 
motor  is  consumed  in  the  resistance  of  the  stator  windings, 
another  portion  is  consumed  in  overcoming  the  stator  iron 
losses,  and  the  remainder  is  supplied  to  the  rotor. 
Expressing  this  statement  in  the  form  of  an  equation,  the 
total  power  supplied  to  the  stator  is 


where  Ph  is  the  sum  of  the  stator  copper  and  iron  losses. 
Similarly,  a  portion  of  the  power  delivered  to  the  rotor  is 
consumed  in  heating  the  rotor  windings,  and  another 
portion,  very  small  and  usually  negligible,  is  consumed  in 
overcoming  the  rotor  iron  losses,  the  rest  being  available 
as  mechanical  power.  A  small  amount  of  the  latter  is 
wasted  in  bearing  friction  and  windage,  but  this  loss  will 
be  ignored.  Then  the  power  input  to  rotor  is 

P    =  P     4-  P 

r  2  <>2      V    r  0> 

where  PC2  is  the  power  expended  in  heating  the  rotor 
windings,  and  where  P0  is  the  mechanical  power  developed 
in  the  rotor.  Combining  these  expressions 

p,  =  pfl  +  pC2  +  P,. 

To  obtain  an  approximate  relation  between  efficiency  and 
speed,  it  is  convenient  to  neglect  the  stator  losses;  that  is, 
the  total  power  taken  by  the  motor  is  considered  as 
effective  in  producing  the  revolving  field.  Then  the 
power  input  to  the  motor  is 

PI  =  P*  =  P,  +  P*. 

If  the  torque  exerted  by  the  rotating  field  upon  the  rotor 
be  T  lb.-ft.,  the  power  required  to  rotate  this  field  at  F 


214          ALTERNATING-CURRENT   MACHINES. 

revolutions  per  minute  against  the  reaction  of  the  rotor 
will  be  2  nVT  ft.-lbs.  per  minute.  The  speed  of  the  rotor 
being  V  revolutions  per  minute,  the  power  developed  in  the 
rotor  will  therefore  be  2  nV'T  ft.-lbs.  per  minute.  The 
power  expended  in  heating  the  rotor  windings  is  2  nVT 
-  2  TiV'T,  or  2  nT  (V  -  F')  ft.-lbs.  per  minute.  Neglect- 
ing the  stator  losses  and  the  rotor  iron  losses,  the  following 
proportion  results: 

P.'.P,:  PC2  =  2  TiVT  :  2  nV'T:  2  nT  (V  -  F'), 
or    Pt  :  P0  :  PC2  =  V  :  V  :  V  -  V  =  i  :  i  -  s  :  s  ; 

from  which 

P        V 
Efficiency  =  j±  =  —  , 

V  =  V  (i  -  s),     P,  =  P2  (i-  s),      and  PC2=  P,s. 

Thus  the  efficiency  of  an  induction  motor  is  approximately 
the  ratio  of  the  rotor  speed  to  the  synchronous  speed, 
and  the  power  expended  in  the  rotor  windings  is  approxi- 
mately proportional  to  the  slip.  Therefore,  to  secure  a 
high  efficiency,  the  slip  should  be  small  so  that  V  more 
nearly  approaches  F;  and  to  have  a  small  slip  requires  a 
rotor  having  windings  of  low  resistance. 

79.  Determination  of  Torque.  —  The  torque  exerted  by 
an  induction  motor  may  be  expressed  in  terms  of  the 
stator  input,  stator  losses,  and  synchronous  speed.  If  P0 
be  the  motor  output  in  watts,  then  the  torque  in  Ibs.  (one 
foot  radius)  is 


2  TtV  746 

But         V  =  V  (i  -  s),     and     P0  =  P2  (i  -  s).      (§  78) 
When  P0  is  expressed  in  watts,  the  term  P2  is  the  rotor 


MOTORS.  215 

input    in    synchronous    watts,    so-called.     Therefore    the 
torque,  neglecting  as  usual  the  rotor  iron  losses,  becomes 

r_  33000*.  (i  -_j)  £. 

27rF(i  -  5)746  V 

The  power  which  is  delivered  to  the  rotor  is  the  differ- 
ence between  the  motor  power  intake  and  the  stator  losses, 

that  is,  P2  *=  P1  —  P;I;  and,  since  V  =  — ^-,    the    expres- 
sion for  torque  becomes 


which  is  independent  of  rotor  speed  and  mechanical  out- 
put. 

80.  The  Transformer  Method  of  Treatment.  —  It  is  cus- 
tomary in  theoretical  discussions  to  consider  the  induction 
motor  as  a  transformer.  Evidently  when  the  rotor  is 
stationary  the  machine  is  nothing  but  a  transformer,  with 
a  magnetic  circuit  so  constructed  as  to  have  considerable 
magnetic  leakage.  When  the  rotor  is  moving,  the  machine 
does  not  act  exactly  like  the  ordinary  transformer,  but  its 
action  can  be  more  conveniently  and  accurately  determined 
by  reference  to  transformer  action.  When  no  load  is  put 
upon  the  rotor  of  an  induction  motor,  the  currents  supplied 
to  the  stator  are  called  the  exciting  currents,  just  as  is  the 
current  in  the  primary  of  a  transformer  when  its  secondary 
winding  is  open-circuited.  The  counter  E.M.F's  induced 
in  the  stator  windings  by  the  revolving  flux  is  less  than  the 
impressed  E.M.F.'s  by  amounts  sufficient  to  allow  the 
exciting  currents  to  flow,  and  these  overcome  the  eddy 
current  and  hysteresis  losses  of  the  stator  iron,  and  set  up 
the  M.M.F.'s  necessary  to  establish  the  rotating  field. 


2l6          ALTERNATING-CURRENT   MACHINES. 

When  the  induction  motor  operates  under  load,  the 
slip,  which  before  was  practically  zero,  is  increased,  and 
E.M.F.'s  are  induced  in  the  rotor  windings  due  to  the 
relative  motion  of  the  rotating  field  and  the  rotor.  The 
demagnetizing  effects  of  the  rotor  currents  produced 
thereby  are  neutralized,  as  in  the  transformer  with  loaded 
secondary,  by  an  increase  of  current  in  the  stator  windings, 
this  being  possible  because  of  the  diminished  counter- 
electromotive  forces.  On  account  of  the  similarity  of  the 
actions  of  the  induction  motor  and  the  transformer,  the 
stator  of  machines  as  ordinarily  constructed  is  also  called 
the  primary,  and  the  rotor  the  secondary  of  an  induction 
motor. 

When  an  induction  motor  is  running  at  a  certain  slip,  s, 
the  frequency  of  the  electromotive  forces  induced  in  the 
rotor  windings  is  s-times  the  frequency  of  the  supply 
voltage.  Because  of  this  fact,  quantities  in  the  secondary 
circuit  cannot  be  directly  added  to  quantities  in  the  pri- 
mary circuit,  but  the  reactions  of  the  rotor  currents  and 
magnetic  fluxes  upon  the  primary  are  of  the  same  fre- 
quency as  the  primary  E.M.F.'s]  for  the  flux  produced  by 
the  secondary  currents  revolves  relative  to  the  rotor  with  a 
speed  equal  to  the  frequency  of  the  induced  secondary 
E.M.F.'s,  so  that  the  speed  of  this  flux  plus  the  speed  of 
the  rotor  is  the  same  as  the  speed  of  the  revolving  field. 
Thus,  the  secondary  flux  of  an  induction  motor  reacts 
upon  the  primary  flux  with  the  same  frequency,  exactly  as 
in  the  transformer. 

81.   Leakage  Reactance  of  Induction  Motors.  —  Not  all 

of  the  flux  set  up  by  the  stator  currents  traverses  the  air 
gap  between  the  rotor  and  stator  iron,  nor  does  all  of  this 


MOTORS. 


217 


air  gap  flux  link  with  the  rotor  turns,  and,  similarly,  not 
all  of  the  flux  set  up  by  the  rotor  currents  links  with  the 
stator  turns.  The  flux  which  links  with  one  winding  and 
not  with  the  other  is  called  the  leakage  flux.  This 
magnetic  leakage  will  be  considered  under  the  following 
heads:  slot  leakage,  tooth-tip  or  " zig-zag"  leakage,  coil- 
end  leakage,  and  belt  leakage. 

Slot  Leakage.     The  flux  which  passes  across  the  slots 
and  the  slot  openings  is  termed  the  slot  leakage  flux,  the 


Fig.  166. 

various  paths  thereof  being  shown  by  the  dotted  lines  in 
Figs.  166  and  167.  The  magnitude  of  the  slot  leakage 
flux  is  dependent  upon  the  form  of  the  slot,  and  may  be 
determined  when  the  dimensions  shown  in  the  figures  are 
known.  Neglecting  the  reluctance  of  the  iron  portion  of 
the  magnetic  circuit,  the  permeance  of  the  path  of  the 
primary  slot  leakage  flux  per  inch  of  slot  length  of  the  slot 
shown  in  Fig.  166  is 


218 


ALTERNATING-CURRENT  MACHINES. 


where  the  subscripts  i  designate  a  primary  slot.  The 
method  of  derivation  of  this  formula  is  identical  with  that 
given  in  §  66. 

The  slot  leakage  flux  per  ampere  inch  of  primary  slot  is 

$S1    =    0.4  T&fa, 

and  hence  the  reactance  of  the  primary  slot  leakage  flux 
per  phase  in  ohms  is 

XSI  =  2  nfls  A/>t  <f>51  io-8, 

where  ls  is  the  length  of  slot  in  inches,  n^  is  the  number  of 
conductors  connected  in  series  per 
primary  slot,  and  Nl  is  the  num- 
ber of  primary  slots  per  phase. 

In  motors  having  full-pitch  wind- 
ings, all  the  conductors  in  one  slot 
belong  to  the  same  phase  wind- 
ing, but  in  motors  having  frac- 
tional-pitch windings  some  or  all 
of  the  slots  contain  conductors 
belonging  to  different  phase  wind- 
ings, and  therefore  the  conductors 
in  these  slots  carry  currents  differing 
in  phase.  To  take  this  influence 
into  account,  the  pitch  factor,  C,  must  be  inserted  in  the 
expression  for  slot  leakage  reactance.  The  value  of  C  is 

plotted   in   terms   of   -     — ^— -  in    Fig.    168.     Then    the 
pole  pitch 

expression  for  the  equivalent  reactance  of  the  primary  slot 
leakage  flux  per  phase  in  ohms  is 


Fig. 167. 


Xtl 


-'•  (I) 


MOTORS. 


219 


Similarly,  the  reactance  of  the  secondary  slot  leakage  flux 
per  phase  in  ohms  is 


GN  \2  p ' 
1T71 )  ^7 ,  where  p^  and  p2  are 
2^V2/    P2 

the  number  of  primary  and  secondary  phases  respectively, 
reduces  the  secondary  slot  leakage  reactance  to  the  primary 
circuit,  or 


s 

^ 

> 

'/ 

y 

i 

//< 

A 

c 

/^ 

/ 

// 

/ 

/ 

C' 

. 

1 

i 

I     . 

)    .( 

• 

i 

}   A. 

COIL  PITCH  v  POLE  PITCH 
Fig.  168. 


If  the  slots  are  of  the  form  shown  in  Fig.  167,  the  per- 
meance of  the  elementary  path  dx  per  inch  of  slot  length  is 


dx 

2.54  — 
y 


2.54 


d  (r  —  r  cos  6) 
—       —  —    - 


r  sin  0 


The  leakage  flux  through  this  element  per  ampere  inch 
of  slot  is 

,,  /         ^N  f    ^y2  -  r  cos  0  .  r  sin  (?1 

d<P4  =  0.4  TT  (1.27  a^)     w ~ I* 

L  Trr  J 


22O 


ALTERNATING-CURRENT   MACHINES. 


Hence  the  inductance  per  phase  (considering  only  the  cir- 
cular portions  of  the  slots)  is 

Ls  =  ^|  n*Nl,  f\0  -  cos  0  sin  0)2  M, 

7T  IO  *J  0 

and  consequently  the  reactance  of  this  portion  of  the  slot 
leakage  per  phase  in  ohms  is 

X.  =  12.5  fn2Nl5io~8  =  2  fl,Nn*  (0.625)  i°~7> 

which  is  similar  to  the  preceding  equations  for  slot  leakage. 
Hence  equations  (i)  and  (2)  may  be  employed  for  calcu- 


Fig.  169. 

lating  the  slot  leakage  reactance  of  round  slots,  when  the 
first  three  terms  within  the  brackets  of  these  expressions 
are  replaced  by  the  constant  0.625. 

Tooth-Tip  Leakage.  Tooth-tip  leakage  is  that  flux  which 
passes  through  a  portion  of  the  tooth  tip  opposite  a  slot. 
The  path  of  this  leakage  flux  is  shown  in  Fig.  169.  For 
convenience  in  the  following  discussion,  the  number  of 
rotor  and  stator  slots  will  be  assumed  equal  and  their 
openings  extremely  small.  The  permeance  of  the  path  of 


MOTORS.  221 


this  leakage  flux  is  variable,  being  zero  when  a  rotor  slot  is 
opposite  a  stator  slot,  and  a  maximum  when  a  rotor  slot  is 
midway  between  two  stator  slots.  The  maximum  per- 
meance per  inch  of  slot  length  is 


where  A  is  the  average  or  common  tooth  pitch,  in  inches, 
and  A  is  the  radial  length  of  the  air  gap,  in  inches.  The 
permeance  of  the  path,  when  the  rotor  and  stator  are  in  an 

intermediate  position,  is  2.54  —  •  —  —  ,  and  it  follows 
that  the  average  permeance  per  inch  of  slot  length  is 

-**)<**  =  0.433  f- 

The  tooth-tip  leakage  flux  per  ampere  inch  of  slot  for 
both  stator  and  rotor  is 

.,  .„  AH* 

<t>,  =  0.4  TrOX  =  0.532  -T*  > 

and  hence  the  equivalent  reactance  of  the  tooth-tip  leakage 
per  phase  (stator  and  rotor)  is 

Xt  =  2  rc/yVX  3>t  io~8. 


When  the  number  of  slots  in  the  rotor  and  in  the  stator 
is  not  the  same,  and  when  the  slot  openings  are  appreci- 

//         /  \2 

able,  the  value  of  &t  must  be  multiplied  by   (  -J-  +  -*  —  i  )  , 

\X1        A2         / 

where  /t  and  t2  are  the  equivalent  stator  and  rotor  tooth- 
tip  widths  respectively,  and  where  A^  and  A2  are  the  stator 
and  rotor  tooth  pitches  respectively.  The  equivalent 


222 


ALTERNATING-CURRENT   MACHINES. 


tooth  tips  are  determined  by  adding  2  A/'  to  the  actual 
tooth-tip  width,  where  /'  is  the  flux-fringing  constant. 
The  value  of  this  constant  is  given  in  Fig.  170,  where/7  is 

plotted  in  terms  of  — j*  •  Then  introducing  the  pitch  fac- 
tor, C,  the  expression  for  the  equivalent  reactance  (stator 
and  rotor)  of  the  tooth-tip  leakage  flux  per  phase  in  ohms  is 


(3) 


For  motors   having  squirrel-cage  rotors,  the  value  of   Xt 
obtained  from  (3)  should  be  reduced  by  20  %. 

Coil-End  Leakage.     The  flux  which  passes  around  the 


~2A 


10 


Fig,  170. 


Fig.  171. 


ends  of  the  coils  where  they  project  beyond  the  slots  is 
called  the  coil-end  leakage  flux,  the  path  of  this  flux  being 
entirely  or  partly  in  air,  as  shown  in  Fig.  171.  It  is 
almost  impossible  to  calculate  accurately  the  coil-end 
leakage  flux  because  of  the  proximity  of  the  motor  end 
plates  and  the  influence  of  the  mutual  flux  of  the  different 
phases.  For  a  full-pitch  three-phase  winding,  it  is  usual 


MOTORS.  223 

to  assume  this  flux  as  one  maxwell  per  ampere  inch  of 
exposed  conductor.  This  assumption  is  experimentally 
justified.  Then  the  flux  in  maxwells  for  all  the  con- 
ductors per  pole  per  phase  (i.e.  per  phase-belt)  per  ampere  is 


where  lc  is  the  length  of  the  end  connections  per  primary 
turn,  and  p  is  the  number  of  pairs  of  poles. 

The  value  of  $>c  depends  upon  the  ratio  of  the  pole  pitch 
to  the  diagonal  of  the  section  of  the  coil  end,  and  is 
approximately  proportional  to  the  logarithm  of  this  ratio. 
But  the  diagonal  of  the  section  of  the  coil  end  is  approxi- 

pole  pitch  Xp    .  ,    . 

mately  equal  to ~  =     ,  , hence  <PC  ispropor- 

number  of  phases      p 

tional  to  log  p'.     For  fractional  pitch  windings,  the  ratio 

7  =  Cr  must  be  introduced.     Therefore  the  value 
pole  pitch 

of  4>c  is  proportional  to  log  C'p'.  The  value  of  log  C'p' 
for  a  full-pitch  three-phase  winding  being  0.477,  tne 
inductance  per  phase-belt  for  any  winding  is  therefore 

0.477 

If  the  length  of  the  rotor  coil-ends  be  considered  80  %  of 
the  stator  coil-ends,  then  the  total  coil-end  inductance  per 
phase  is 

LK  =  *.' 


and  the  coil-end  leakage  reactance  per  phase  (stator  and 
rotor)  in  ohms  is 

X,  =  5-95  /  ^^  I.  log  C'p' .  io-8.  (4) 


224 


ALTERNATING-CURRENT   MACHINES. 


Belt  Leakage.  Neglecting  the  exciting  current  of  the 
induction  motor,  and  considering  an  instant  when  a 
primary  phase-belt  completely  overlaps  a  secondary  phase- 
belt,  the  magnetomotive  forces  due  to  the  currents  in  these 
belts  of  conductors  will  be  in  opposition,  and  there  will  be 
no  belt  leakage.  But,  if  the  belts  of  conductors  be  in 


Fig.  172. 

any  other  position,  a  secondary  phase-belt  overlaps  two 
primary  phase-belts,  and  their  magnetomotive  forces  are 
no  longer  in  opposition.  The  resultant  M.M.F.  will  be 
effective  in  producing  a  leakage  flux,  termed  belt  leakage, 
which  links  with  primary  and  secondary  conductors,  thus 
resulting  in  a  leakage  reactance.  The  relative  position  of 
stator  and  rotor  for  which  this  reactance  is  a  maximum,  is 
shown  in  Fig.  172,  the  paths  of  the  flux  being  indicated  by 
the  dotted  lines,  and  the  slots  for  conductors  of  different 
phases  being  lettered  differently. 

An  expression  for  the  average  belt  leakage  inductance 
per  phase,  similar  to  that  given  by  Adams,  is 


Lb  = 


MOTORS. 


225 


where  512  is  the  number  of  series  conductors  per  phase  per 
pole  (primary  and  secondary),  D  is  the  rotor  diameter  in 
inches,  k'  is  3.32  for  two-phase  and  1.005  f°r  three-phase 


motors,  K  is  the  slot  contraction  factor,  or  -^  ,  Kl  is  a  con-' 

\*2 

stant  depending  upon  the  number  of  slots  per  pole  as 
obtained  from  Fig.  173,  K2  is  a  constant  taking  into 
account  the  ampere  turns  for  the.  iron  portions  of  the 


16 

SLOTS  PER  POLE 

Fig-  173- 


belt  leakage  paths  and  may  be  taken  as  0.85,  and  C  is 
the  pitch  factor  as  determined  from  Fig.  168.  The  belt 
leakage  reactance  per  phase  (primary  and  secondary),  in 
ohms,  is  2  7ifLb)  or 


Xb  = 


(5) 


where  k"  is  17.8  for  two-phase  and  5.36  for  three-phase 
motors.  This  expression  applies  to  induction  motors 
having  phase-wound  rotors;  for  motors  having  squirrel- 
cage  rotors  the  value  of  k"  should  be  reduced  by  about 

65  %• 


226 


ALTERNATING-CURRENT    MACHINES. 


Total  Leakage  Reactance.  The  total  leakage  reactance 
per  phase  of  an  induction  motor  is  the  sum  of  the  various 
leakage  reactances  for  which  expressions  have  just  been 
derived.  That  is,  the  total  reactance  per  phase,  XT,  is 
equal  to  the  sum  of  equations  (i),  (2),  (3),  (4),  and  (5). 

To  secure  a  high  starting  torque  and  efficiency  it  is 


\ 


V 


ST.    -90          80          70         60          50         40          30         20          10        6YN. 
PER  CENT.  SLIP 

Fig.  174. 

necessary  to  keep  the  magnetic  leakage  as  small  as  possi- 
ble. The  relation  of  torque  to  speed  with  various  arbi- 
trary values  of  magnetic  leakage  is  shown  in  Fig.  174.  It 
is  seen  that  the  maximum  torque  increases  directly  as  the 
leakage  decreases. 

The  leakage  reactance  of  induction  motors  may  be 
decreased  by  employing  fractional-pitch  windings,  and  by 
increasing  the  reluctance  of  the  path  of  the  leakage  flux. 
The  reluctance  of  the  path  of  the  useful  flux,  however, 


MOTORS.  227 

should  be  kept  as  low  as  possible,  and  it  is  usual  to  make 
the  air  gap  just  as  small  as  is  consistent  with  good  mechani- 
cal clearance.  Concentricity  of  rotor  and  stator  is  to  be 
obtained  by  making  the  bearings  in  the  form  of  end  plates 
fastened  to  the  stator  frame.  Some  makers  send  wedge 
gap-gauges  with  their  machines  so  that  a  customer  may 
test  for  eccentricity  due  to  wear  of  the  bearings.  A  small 
air  gap,  besides  lowering  the  leakage  and  raising  the  power 
factor,  increases  the  efficiency  and  capacity  of  the  motor. 

A  convenient  expression  giving  the  proper  radial  depth 
of  the  air  gap  in  inches,  in  terms  of  the  horsepower  rating 
of  the  motor,  is 


82.  Calculation  of  Exciting  Current.  —  The  exciting 
current  per  phase  of  an  induction  motor  has  two  com- 
ponents; one,  the  power  component,  which  overcomes  the 
eddy  current  and  the  hysteresis  losses  of  the  iron,  the 
small  stator  copper  loss  due  to  the  exciting  current  being 
neglected;  and  the  other,  the  wattless  component  of  the 
exciting  current,  which  sets  up  the  magnetomotive  force 
necessary  to  overcome  the  reluctance  of  the  magnetic 
circuit.  The  iron  losses  of  the  rotor  under  normal  con- 
ditions are  extremely,  small  because  of  the  very  low  fre- 
quency of  rotor  flux;  therefore  only  the  stator  iron  losses 
need  be  considered.  In  the  following  discussion,  star- 
connected  windings  are  assumed. 

The  power  component  of  the  exciting  current  per  phase  is 


228  ALTERNATING-CURRENT   MACHINES. 

where  Pe  is  the  total  eddy  current  loss,  Ph  is  the  total 
hysteresis  loss,  E  is  the  impressed  electromotive  force  per 
stator  winding,  and  p'  is  the  number  of  phases.  The 
values  of  Pe  and  Ph  must  be  calculated  separately  for 
the  stator  teeth  and  for  the  stator  yoke,  because  the  flux 
densities  in  these  parts  of  the  magnetic  circuit  are  different. 
The  maximum  flux  density  in  the  teeth  will  first  be 
determined. 

The  counter  E.M.F.  induced  in  a  full-pitch  distributed 
stator  winding  (one  phase)  by  the  rotating  magnetic  field  is 

ik&pQS^-io-*,  §42 

oo 

where  k^  is  the  form  factor  or  the  ratio  of  the  effective  to 
the  average  E.M.F.,  k2  is  the  distribution  constant  as 
obtained  from  Fig.  57,  p  is  the  number  of  pairs  of  stator 
poles,  <l>  is  the  flux  per  pole  and  is  assumed  to  be  sinu- 
soidally  distributed,  5  is  the  number  of  conductors  con- 
nected in  series  per  stator  phase,  and  V  is  the  rotor  speed 
in  revolutions  per  minute.  In  single-phase  motors,  the 
winding  is  usually  distributed  over  two-thirds  of  the  pole 
distance.  When  the  rotor  revolves  at  synchronous  speed, 

V 

then  p  —  =  /.     If   the   ohmic   drop   due   to   the  exciting 
60 

current  in  the  stator  winding  be  neglected,   the  counter 
E.M.F.  and  the  impressed  E.M.F.  are  practically  equal. 
Finally,  introducing  the  pitch  factor,  C,  as  obtained  from 
Fig.  168,  to  take  care  of  fractional-pitch  windings,  the  value 
of  the  impressed  electromotive  force  becomes 
E  =  2&1&2C<J>S/io-8, 
Eio* 

whence  ** 


MOTORS.  229 

Representing  the  core  length  in  inches  by  ls,  and  the  pole 
pitch  in  inches  by  Xp,  the  average  flux  density  (maxwells 
per  square  inch)  in  the  air  section  becomes 


and  the  maximum  flux  density 


2   ;      2 1SXP 

This  equation  assumes  a  continuous  surface  on  both 
sides  of  the  air  gap  over  the  polar  region,  but  this  does  not 
occur  in  practice  because  of  the  presence  of  rotor  and 
stator  slots.  In  the  following,  the  rotor  slots  are  assumed 
extremely  small  so  that  their  influence  on  flux  distribution 
is  inappreciable.  If  t±  be  the  equivalent  stator  tooth  tip 
width  in  inches,  and  ^  be  the  stator  tooth  pitch  in  inches, 
then  the  maximum  flux  density  (maxwells  per  square  inch) 
in  the  air  gap  as  well  as  in  the  tooth,  which  at  that  instant 
is  at  the  center  of  the  polar  region,  is 

««     =    S-     =  ^'£l°8 


4 

which  is  the  value  to  be  taken  for  the  maximum  flux 
density  in  calculating  the  eddy  current  and  hysteresis 
losses  in  the  stator  teeth. 

The  maximum  flux  density  in  the  stator  yoke  is  half  of 
the  maximum  flux  density  in  the  air  gap  where  con- 
tinuous surfaces  on  both  rotor  and  stator  were  assumed. 
Therefore  the  maximum  flux  density  to  be  employed  in 
calculating  the  eddy  current  and  hysteresis  losses  in  the 
stator  yoke  is 

'  8  , 


230  ALTERNATING-CURRENT   MACHINES. 

The  values  of  Pe  and  Ph  are  calculated  as  in  §  61. 

The  wattless  component  of  the  exciting  current,  or  the 
magnetizing  current,  of  an  induction  motor  supplies  the 
M.M.F.  necessary  to  overcome  the  air-gap  reluctance, 
the  reluctance  of  the  iron  being  neglected.  The  magneto- 
motive force  required  is 


M.M.F.  = 


2.54 


where  A  is  the  radial  length  of  the  air  gap  in  inches. 

In  a  two-phase  machine,  the  magnetizing  currents  in 
both  phases  together  set  up  this  magnetomotive  force. 
When  the  magnetizing  current  in  one  winding  is  at  maxi- 
mum value  (\/r2  Imag),  at  that  instant  the  magnetizing 
current  in  the  other  winding  is  zero.  Therefore  .the  total 
M.M.F.  set  up  per  pole  per  phase  will  be 

M.M.F.- 


10  .  2  p 

Equating  (3)  and  (4)  and  solving,  the  magnetizing  current 
per  phase  for  a  two-phase  induction  motor  is 

g 


5.08  \/2   k&Cl^t^f 

In  a  three-phase  machine,  at  the  instant  when  the 
magnetizing  current  in  one  winding  is  at  maximum  value 
(\/2  I  mag),  the  magnetizing  current  in  each  of  the  other 

two  windings  is  at  half  maximum  value  (  -  Imag  ),  and 
hence  the  current  which  sets  up  the  M.M.F.  is  \/2lmag 
+  2  f  —  -  Imag  J=  2  V~2  Imag,  and  the  magnetomotive  force 


MOTORS.  231 

supplied  thereby,  per  pole  per  phase,  is 


M.M.F.  =OTq,. 

10  .  2  p 

Equating  (3)  and  (6)  and  solving,  the  magnetizing  current 
per  phase  for  a  three-phase  induction  motor  is  found  to  be 
one-half  that  for  a  two-phase  motor,  or  half  that  given  by 
equation  (5).  It  should  be  noted  that  in  the  foregoing, 
E  is  the  E.M.F.  impressed  upon  each  stator  winding,  and 
not  the  E.M.F.  across  motor  terminals. 

After  the  two  components  of  the  exciting  current  have 
been  calculated,  the  magnitude  of  Iexc  per  phase  may  be 
obtained  from  the  relation 

/«,  =  ^Un  +  /,4  >  (7) 

and  its  angle  of  lag  behind  the  impressed  E.M.F.  is  given 
by 

&  =  cot-'--  §62 


In  induction  motors  the  size  of  Ie+h  is  small  compared  to 
Imag,  the  exciting  current  differing  from  the  magnetizing 
current  by  but  a  few  per  cent. 

83.  Circle  Diagram  by  Calculation.  —  The  similarity 
between  an  induction  motor  operating  under  a  mechanical 
load  and  a  transformer  operating  under  a  resistance  load 
has  already  been  pointed  out;  from  whence  it  follows  that 
the  transformer  circle  diagram,  §  68,  may  be  applied  to 
the  induction  motor.  The  circle  diagram  may  be  con- 
structed when  the  magnitude  and  phase  of  the  exciting 
current  are  known,  and  when  the  leakage  reactance  of  the 
motor  has  been  calculated.  The  magnitude  and  position 


232          ALTERNATING-CURRENT    MACHINES. 

of  the  exciting  current  per  phase  may  be  computed  from 
the  expressions  derived  in  the  preceding  article.  This 
value  is  laid  off  as  in  Fig.  175  and  the  point  D  is  thus 
located,  which  is  one  of  the  points  on  the  circular  current 
locus.  The  diameter  of  this  circle  is  the  ratio  of  the 
impressed  E.M.F.  per  stator  winding  to  the  total  leakage 

Tf 

reactance  per  phase.      That  is,  DC  =  —  >  where  XT  is 

XT 

the  sum  of  equations  (i),  (2),  (3),  (4),  and  (5)  of  §  81. 
Thus  the  circle  diagram  is  completely  determined. 

The  current  in  a  stator  winding  of  an  induction  motor 


WATTLESS  CURRENT 

Fig.  175- 


may  be  considered  as  the  resultant  of  two  components :  one, 
the  primary  exciting  current  per  phase,  and  the  other,  the 
effective  current  which  supplies  the  magnetomotive  force 
necessary  to  counterbalance  the  M.M.F.  per  phase  due  to 
the  rotor  currents.  Thus  in  Fig.  175,  OG  is  the  stator 
current  per  phase,  and  is  the  resultant  of  OD,  the  exciting 
current  per  phase,  and  DG,  the  effective  current  per  phase. 
The  power  factor  of  an  induction  motor  depends  upon 
the  value  of  the  stator  current,  as  may  be  seen  from  the 


MOTORS.  233 

figure.  The  maximum  power  factor  is  attained  when  OG 
is  tangent  to  the  semicircle.  Neglecting  the  power  com- 
ponent of  the  exciting  current,  DB,  the  maximum  power 
factor  may  be  expressed  as 

DC  DC 

,                              2                                      2  I 

COS  (f>m  =     — -   = — -  -  > 


AD 

where  <r  =  ——  =  leakage  coefficient.     The  determination 
u\^ 

of  the  performance  curves  of  an  induction  motor  from  the 
circle  diagram  will  be  considered  later. 

84.  Circle  Diagram  by  Test.  —  The  circle  diagram  of 
an  induction  motor  is  completely  determined  and  may  be 
constructed  when  the  magnitude  and  position  of  the 
exciting  current  per  phase  and  when  the  magnitude  and 
corresponding  position  of  any  other  current  value  per 
stator  phase  are  known. 

The  exciting  current  per  stator  phase  can  be  determined 
when  the  primary  amperes,  the  primary  voltage  between 
line  wires,  and  the  watts  input  have  been  obtained  while 
the  motor  operates  at  no-load  with  full  voltage.  Thus, 
for  a  three-phase  induction  motor  having  its  stator  wind- 
ings Y-connected,  Fig.  176,  the  magnitude  of  Iexc  per 
phase  is  the  ammeter  reading.  The  power  input  is  the 
sum  of  the  wattmeter  readings  in  the  two  positions,  that  is, 
pi  +  P2  =  ^EJesc  cos  fa.  §  47 

Therefore    the    angle    by  which  Iexc  lags   behind   the  im- 
pressed E.M.F.  is 

,  t    P.  +  P. 

fa  =   CO"1 


234 


ALTERNATING-CURRENT   MACHINES. 


where  Et  is  the  voltmeter  reading.  The  exciting  current 
per  phase,  OD,  Fig.  175,  is  now  completely  determined  and 
may  be  drawn  to  a  convenient  scale.  The  point  D  con- 
stitutes one  point  on  the  circular  current  locus. 

If  current,  voltage,  and  power  measurements  be  made 
when  the  motor  is  operating  under  load,  or  when  the  rotor 
is  locked,  another  point,  G,  on  the  current  locus  may  be 


Fig   176. 


similarly  located.  The  latter  measurement  is  to  be  pre- 
ferred because  it  determines  the  extreme  value  of  the 
stator  current,  and  intermediate  points  on  the  diagram  are 
less  likely  to  be  in  error.  The  impressed  E.M.F.,  for  the 
measurement  with  rotor  locked,  should  be  reduced  to 
such  a  value  as  will  send  a  current  which  will  produce  the 
same  heating  in  the  windings  as  does  continuous  full-load 
operation.  The  current  and  power  may  then  be  calculated 
for  full  voltage  by  increasing  the  former  in  direct  pro- 
portion to  the  voltage,  and  by  increasing  the  latter  in 
proportion  to  the  square  of  the  voltage.  The  corrected 
current  value  is  then  laid  off  on  the  diagram  to  the  same 
scale.  The  semicircle  drawn  through  the  points  D  and 


MOTORS.  235 

G,  having  a  center  on  a  line  through  D  perpendicular  to 
the  E.M.F.  vector,  is  the  required  current  locus. 

Having  constructed  the  circle  diagram,  the  total  leakage 
reactance  per  phase  of  the  induction  motor  may  be  readily 
obtained,  since  the  diameter  of  the  semicircle  is  the  ratio 
of  the  volts  per  stator  phase  to  the  total  reactance  per 

phase.     The   volts   per   stator   phase   in   the   three-phase 

•p 
induction  motor  with  Y-connected  windings  is  — zr  • 

^3 

Before  proceeding  to  the  calculation  of  the  performance 
curves  of  an  induction  motor,  it  is  necessary  to  know  the 
resistance  per  phase  of  the  stator  winding  and  the  equiva- 
lent rotor  resistance  per  stator  phase.  The  resistance,  Rlt 
between  terminals  of  the  stator  winding  may  be  directly 
measured  by  direct-current  methods;  thus  for  a  three- 
phase  motor,  the  total  stator  copper  loss  (§  50)  is  f  PRV 
where  /  is  the  current  per  line;  hence  the  stator  copper 

PR 
loss  per  phase  is  . — -,  and  the  resistance  per   phase  is 

R. 
V:t" 

The  equivalent  rotor  resistance  per  stator  phase,  r2,  of  a 
three-phase  coil-wound  rotor  with  a  winding  identical  with 
that  of  the  stator,  is  half  the  rotor  resistance  measured 
between  two  slip-rings.  Therefore  the  copper  loss  of  this 
coil-wound  rotor  per  stator  phase  is  Prr 

The  equivalent  resistance  of  a  squirrel-cage  rotor  per 
stator  phase  cannot  be  determined  directly,  but  may  be 
calculated  from  the  observations  taken  during  the  motor 
test  with  the  rotor  locked.  As  already  explained,  the 
impressed  E.M.F.  for  this  test  is  reduced,  and  the  current 
and  power  input  must  thereafter  be  computed  for  full 


236 


ALTERNATING-CURRENT    MACHINES. 


voltage  operation.  Representing  the  stator  current  per 
phase  at  lock  by  //,,  the  power  per  phase  at  lock  by  PL) 
and  the  constant  iron  and  friction  loss  per  phase,  as 
obtained  at  no-load  running,  by  Pfi,  then  the  copper  loss 
in  the  squirrel-cage  rotor  of  an  induction  motor  per  stator 
phase  is 


and  the  equivalent  resistance  of  the  squirrel-cage  rotor  per 
stator  phase  is 


85.   Performance  Curves  from  Circle  Diagram.  —  Having 
constructed   the  circle  diagram  as  described  in   the  two 


Fig.  177. 


foregoing  articles,  the  performance  curves  of  the  induction 
motor  may  be  determined  therefrom  by  calculating  the  fol- 
lowing quantities  for  a  number  of  positions  of  the  point  G 
on  the  current  locus,  Fig.  177. 


MOTORS.  237 

Stator  Current  per  phase  =  OG. 

Input  to  Motor  =  E  .  OK  .  pr  . 

OK 
Power  Factor  of  Motor  =  cos  <j>  =  —  — 

OG 

Stator  Copper  Loss  =  rt  .  OG2  .  p'. 

Input  to  Rotor  =  E  .  AK  .  p'  -  r^  .  OG2  .  p'. 
Rotor  Copper  Losses  =  r2  .  DG*  .  p'. 
Motor  Mechanical  Output  = 

pf  [E.AK-r1.OG2-r2.  ~DG2]. 

„  ~  .  Mechanical  Output 

Motor  Efficiency  =  E    OK    p'        ' 


slip  =  —  *-        ^  •      §  78 

E.AK  -  r^OG 

E.AK.p'-r,.O(?.p' 
Torque  =  c.i  174  p—  —  —     -  *—•     §79 

In  the  foregoing,  E  is  the  impressed  E.M.F.  per  phase, 
p'  is  the  number  of  phases,  rt  is  the  stator  resistance  per 
phase,  r2  is  the  equivalent  rotor  resistance  per  stator  phase, 
p  is  the  number  of  pairs  of  poles,  and  /  is  the  frequency. 

The  results  obtained  may  then  be  embodied  in  a  set  of 
curves  as  in  Fig.  178,  where  abscissae  represent  per  cent 
full-load  power  output,  and  ordinates  represent  per  cents. 

If  the  voltage  impressed  upon  an  induction  motor  be 
increased,  there  will  result  a  proportional  increase  in  the 
flux  linked  with  the  rotor,  and  in  consequence  a  propor- 
tional increase  in  the  rotor  current.  As  the  torque  de- 
pends upon  the  product  of  the  flux  and  the  rotor  ampere 
turns,  it  follows  that  the  torque  varies  as  the  square  of  the 
impressed  voltage.  The  capacity  of  a  motor  is  therefore 


ALTERNATING-CURRENT   MACHINES. 


changed  when  it  is  operated  on  circuits  of  different  volt- 
ages. 

Owing  to  the  low  power  factor  of  induction  motors, 
transformers  intended  to  supply  current  for  their  operation 
should  have  a  higher  rated  capacity  than  that  of  the 


40  60  80  100  120 

PERCENT  FULL-LOAD  POWER  OUTPUT 


motors.  It  is  customary  to  have  the  kilowatt  capacity  of 
the  transformer  equal  to  the  horsepower  capacity  of  the 
motor. 

The  direction  of  rotation  of  a  three-phase  motor  can  be 
changed  by  transposing  the  supply  connections  to  any  two 
terminals  of  the  motor.  In  the  case  of  a  two-phase,  four- 
wire  motor,  the  connections  to  either  one  of  the  phases 
may  be  transposed. 


86.   Method   of  Test  with  Load.  —  The   complete   per- 
formance of  two-phase  or  three-phase  induction   motors 


MOTORS.  239 

when  operated  from  balanced  two-phase  or  three-phase 
circuits  may  be  calculated  when  the  values  of  power  input, 
as  measured  by  the  two-wattmeter  method,  §  47,  have  been 
determined  by  test  for  various  mechanical  loads  upon  the 
rotor.  The  instruments  required  are  a  voltmeter  and  a 
wattmeter,  three  observations  being  necessary  at  each 
load,  namely,  Pv  P2,  and  the  line  voltage,  Et.  The  pri- 
mary resistance  measured  between  terminals  and  the 
equivalent  secondary  resistance  per  stator  phase  must  be 
known.  An  outline  of  the  method  employed  for  a  three- 
phase  induction  motor  follows. 
By  reference  to  §  47,  it  is  seen  that 

Total  primary  input  =  Pl  +  P2  =  \/3  EJ  cos  <£,  (i) 

P3  -  Pt  =  EJ  sin  </>,  and  (2) 

[p    p  -i 
V^   p2       p1  I  •  (3 ) 
*|      '      ^2-J 

The  primary  current  per  terminal,  as  obtained  from  (2),  is 
P2  -  pt 

_T      _p    _p-i 

where  sin  ©  =  sin  tan       vT  —2 *-*  •  (e\ 

P    -4-   P    I 

The  equivalent  single-phase  current  is  V$  /,  §50.  The 
power  and  wattless  components  of  the  equivalent  single- 
phase  current  are  respectively 

(6) 


and  V^  /  sin  <j)  =  \/3  — ? ^  •  (7) 

The  primary  copper  loss,  §  50,  is  |  PRV  (8) 


240          ALTERNATING-CURRENT   MACHINES. 

where  R^  is  the  stator  resistance  measured  between  ter- 
minals. The  value  of  /  is  given  by  (4)  and  (5). 

The  iron  and  friction  loss  is  obtained  by  subtracting  the 
primary  copper  loss  at  no-load  from  the  total  primary 
input  at  no-load,  that  is, 

Pj.-JV  +  P/-*/.'^,  (9) 

where  P*,  P2°,  and  70  are  the  no-load  values.  70  is 
obtained  from  (4)  and  (5)  by  taking  P2°  for  P2,  and  P* 
for  Pr 

The  total  primary  losses  including  friction  are  therefore 

Pfi  +  I  PR,.  (10) 

The  secondary  input  in  synchronous  watts  is  the  differ- 
ence between  the  total  primary  input  and  the  total  primary 
losses,  or 

P.-.-P.  +  P.-P,-  JP*,.  (n). 

The  external  rotor  torque,  §79,  is  0.1174  —T~>          (12) 

where  p  is  the  number  of  pairs  of  poles  and  /  is  the  fre- 
quency of  the  impressed  E.M.F. 

The  power  and  wattless  components  of  the  equivalent 
secondary  current  are  respectively 

/2  cos  &  =  (6  )  -  (6  at  no-load  )  =  P*  +  PZ  ~J°  ~  P*>    (13) 

j&z 

and 


hence  the  equivalent  single-phase  secondary  current  is 


I2=  V  (7a  cos  02)2  +  (/2  sin  <£2)2.  (15) 


MOTORS.  241 

The  secondary  copper  loss  =  r2/22,  where  r2  is  the  equiv- 
alent secondary  resistance. 

The  percentage  rotor  slip  is  the  ratio  of  the  secondary 
copper  loss  to  the  secondary  input  in  synchronous  watts,  or 

100  r,/,2  .  ,N 

*  =       p2  *  •  (l6) 

rr 

The  output  of  the  motor  in  horse  power  is 
P  —  r  1  2 

^  ft« 

The  efficiency  of  the  induction  motor,  being  the  ratio  of 
the  watts  output  to  the  watts  input,  is 

Eff-  =  '''  (I8) 


which,    when    multiplied    by    100,    gives    the    percentage 
efficiency. 

When  numerous  values  of  P1  and  P2  have  been  experi- 
mentally determined  and  when  the  foregoing  computations 
have  been  made  for  each  set  of  readings,  curves  of  the 
various  factors  may  be  plotted  in  terms  of  the  motor  out- 
put, as  in  the  preceding  article. 

87.  Phase  Splitters.  —  In  order  to  operate  polyphase 
induction  motors  upon  single-phase  circuits,  use  is  made  of 
inductances  in  series  with  one  motor  circuit  to  produce  a 
lagging  current,  or  of  condensers  to  produce  a  leading  cur- 
rent, or  of  both  —  one  in  each  of  two  legs.  The  General 
Electric  Company,  in  its  condenser  compensator,  for  use 
with  small  motors,  as  shown  in  Fig.  179,  employs  an 
autotransformer  and  condenser,  connected  as  in  diagram, 
Fig.  1  80. 


242  ALTERNATING-CURRENT   MACHINES. 


The   autotransformer   is   used    to    step-up  the  voltage, 
which  is    impressed    upon    the    condenser,    to    500    volts. 


Fig.  179. 

The  necessary  size  of  the  condenser  is  thereby  reduced. 

The  equivalent  impedance  of  the  autotransformer  and 
condenser,  as  connected,  is  such  as  to  pro- 
duce a  leading  current  in  the  one-phase 
sufficient  to  give  a  satisfactory  starting 
torque,  and  it  brings  the  power  factor  prac- 
tically up  to  unity  at  all  loads. 


88.   Single-Phase    Induction    Motors.  — 

The  difference  between  the  single-phase 
and  the  polyphase  induction  motor  lies 
principally  in  the  character  of  their  mag- 
netic fields.  In  the  polyphase  motor,  the 
revolving  field  is  practically  sinuscidally 
distributed  in  space  and  constant  in  value. 
This  is  also  true  of  the  single-phase  induc- 
tion motor  when  running  at  synchronous  speed,  but  at  any 
other  speed  the  field  is  not  constant  in  value,  nor  is  it 


Fig.  180. 


MOTORS.  243 

sinusoidally  distributed  in  space.  If  an  alternating  E.M.F. 
be  impressed  upon  the  stator  winding  of  a  single-phase 
induction  motor  an  alternating  flux  will  be  set  up  which 
passes  through  the  rotor.  This  pulsating  flux  lags  approxi- 
mately 90°  behind  the  impressed  E.M.F.  When  the  rotor 
is  in  motion,  its  conductors  cut  this  flux  and  E.M.F. 's  are  set 
up  in  them  which  are  in  time  phase  with  the  flux.  The  value 
of  these  generated  E.M.F. 's  depends  upon  the  magnitude 
of  the  stator  flux  and  upon  the  speed  of  the  rotor.  They  set 
up  currents  in  the  rotor  conductors,  the  magnitudes  of  which 
are  directly  proportional  to  the  electromotive  forces  generated 
therein.  These  currents  set  up  a  rotor  flux  which  lags  90° 
in  time  behind  the  rotor  E.M.F. ,'s,  and  is  displaced  90  elec- 
trical degrees  in  space.  Thus  the  pulsating  flux  through  the 
rotor  due  to  the  impressed  alternating  E.M.F.  is  at  right 
angles,  in  a  bipolar  machine,  to  the  rotor  flux  due  to  the 
motion  of  the  rotor.  When  the  stator  flux  is  a  maximum 
there  will  be  no  rotor  flux,  and  when  the  stator  flux  is  zero 
the  rotor  flux  will  be  a  maximum.  In  this  way  the  resultant 
magnetic  flux  in  the  gap  changes  its  position  and  revolves 
in  the  direction  of  rotation.  At  synchronous  speed,  the 
maximum  values  of  the  stator  flux  and  the  rotor  flux  are 
equal;  thus  a  true  rotating  field  of  uniform  intensity  is  pro- 
duced. At  a  lower  speed,  the  maximum  values  are  unequal 
and  consequently  the  rotating  field  will  be  of  variable  inten- 
sity. At  standstill  no  revolving  field  exists  and  no  torque  is 
developed. 

The  inability  of  the  single-phase  induction  motor  to 
exert  a  torque  at  standstill  has  led  to  the  introduction  of 
numerous  starting  devices,  but  these  are  usually  only 
applicable  to  small-sized  motors.  Three  general  methods 


244          ALTERNATING-CURRENT   MACHINES. 

are  employed  to  render  the  motor  self-starting  under  load. 
First,  the  motor  can  be  started  as  a  repulsion  motor 
(§103),  and  when  normal  speed  is  attained,  a  centrifugal 
device  automatically  short-circuits  the  commutator  and 
simultaneously  lifts  off  the  brushes,  thus  changing  the 
machine  to  a  single-phase  induction  motor.  Second, 
an  auxiliary  stator  winding  may  be  connected  to  the  line 
through  a  phase-splitting  device,  as  in  §  87.  Either  a 
squirrel-cage  or  a  coil-wound  rotor  may  be  used.  Third, 
an  auxiliary  winding  on  the  stator  is  connected  through  a 
non-inductive  resistance  and  switching  device  to  the  line. 
An  automatic  clutch  is  employed,  thus  permitting  the  motor 
to  approach  normal  speed  before  taking  up  its  load. 

89.  The  Monocyclic  System.  —  This  is  a  system  advo- 
cated by  the  General    Electric    Company  for   the    use  of 
plants   whose   load   is   chiefly  lights,   but   which   contains 
some   motors.     The   monocyclic   generator  is   a   modified 
single-phase  alternator.     In  addition  to  its  regular  winding, 
it  has  a  so-called  teaser  winding,  made  of  wire  of  suitable 
cross-section  to  carry  the   motor  load,  and  with    enough 
turns  to  produce  a  voltage  one-fourth  that  of  the  regular 
winding,  and  lagging  90°  in  phase  behind  it.     One  end  of 
the  teaser  winding  is  connected  to  the  middle  of  the  regu- 
lar winding,  and  the  other  end  is  connected  through  a  slip- 
ring  to  a  third  line  wire. 

A  three-terminal  induction  motor  is  used,  the  terminals 
being  connected  to  the  line  wires  either  directly  or  through 
transformers. 

90.  Frequency  Changers.  —  These  are  machines  which 
are  used  to  transform  alternating  currents  of  one  frequency 


MOTORS.  245 

into  those  of  another  frequency.  They  are  commonly  used 
to  transform  from  a  low  frequency  (say  from  25  or  40)  to 
a  higher  one.  They  depend  for  their  operation  upon  the 
variation  with  slip  of  the  frequency  of  the  rotor  E.M.F.'s 
of  an  induction  motor.  The  common  practice  for  raising 
the  frequency  is  to  have  a  synchronous  motor  turn  the 
rotor  of  an  induction  motor  in  a  direction  opposite  to  the 
direction  of  rotation  of  the  latter's  field.  The  synchronous 
motor  and  the  stator  windings  of  the  induction  motor  are 
connected  to  the  low-frequency  supply  mains.  Slip-rings 
connected  to  the  rotor  windings  of  the  induction  motor 
supply  current  at  the  higher  frequency.  The  size  of  the 
synchronous  motor  necessary  to  drive  the  frequency 
changer  is  the  same  percentage  of  the  total  output  as  the 
rise  of  frequency  is  to  the  higher  frequency. 

91.  Speed  Regulation  of  Induction  Motors.  —  The  speed 
of  an  induction  motor  can  be  varied  by  altering  the  voltage 
impressed  upon  the  stator,  by  altering  the  resistance  of  the 
rotor  circuit,  or  by  commutating  the  stator  windings  so  as 
to  alter  the  multipolarity.  The  first  two  methods  depend 
for  their  operation  upon  the  fact  that,  inasmuch  as  the 
motor  torque  is  proportional  to  the  product  of  the  stator 
flux  and  the  rotor  current,  for  a  given  torque  the  product 
must  be  constant.  Lessening  the  voltage  impressed  upon 
the  stator  lessens  the  flux,  and  also  the  rotor  current,  if  the 
same  speed  be  maintained.  The  speed,  therefore,  drops 
until  enough  E.M.F.  is  developed  to  send  sufficient  current 
to  produce,  in  combination  with  the  reduced  flux,  the 
equivalent  torque.  Increasing  the  resistance  of  the  rotor 
circuit  decreases  the  rotor  current,  and  requires  a  drop  in 
speed  to  restore  its  value.  Both  of  these  methods  result 


246  ALTERNATING-CURRENT    MACHINES. 

in  inefficient  operation.  If  the  impressed  voltage  be 
reduced,  the  capacity  of  the  motor  is  reduced.  In  fact,  the 
capacity  varies  as  the  square  of  the  impressed  voltage. 
Changes  in  the  multipolarity  of  the  stator  require  compli- 
cated commutating  devices. 

92.  The  Induction  Wattmeter.  —  The  operation  of  the  in- 
duction wattmeter,  like  that  of  the  induction  motor,  is  based 


Fig.  181. 

upon  the  action  of  a  revolving  or  shifting  magnetic  field 
upon  a  metallic  body  capable  of  rotation.  The  rotating 
field  is  developed  because  of  the  difference  in  phase  of  the 
magnetic  fields  produced  by  the  currents  in  the  series  and 
shunt  coils  of  the  wattmeter.  The  coils  and  the  rotating 
member  of  an  induction  wattmeter  are  shown  assembled  in 
Fig.  181.  The  disk  or  armature  is  carried  on  a  short 


MOTORS.  247 

shaft  which  is  mounted  in  the  usual  way  and  is  provided 
with  a  worm  gear  at  its  upper  end  for  actuating  the  dial 
train. 

The  series  coil  has  no  iron  core  and  consists  of  a  few 
turns  of  heavy  wire,  thus  it  possesses  very  little  self-induction. 
If  the  power  factor  of  the  load  circuit  be  unity,  then  the 
current  flowing  through  this  winding  will  be  practically  in 
phase  with  the  impressed  E.M.F.,  and,  as  the  flux  is  in 
phase  with  the  current  producing  it,  this  also  will  be 
approximately  in  phase  with  the  impressed  electromotive 
force.  The  shunt  coil  consists  of  a  large  number  of  turns 
of  small  wire  wound  on  a  laminated  iron  core.  This 
winding  has  considerable  self  induction,  hence  the  current 
flowing  through  it  is  almost  at  right  angles  to  the  impressed 
E.M.F.  This  angle  of  lag  is  slightly  less  than  90°  owing 
to  the  iron  and  copper  losses  of  the  shunt  circuit.  The 
vector  diagram  corresponding  to  these  conditions  is  shown 
in  Fig.  182. 

The  alternating  magnetic  fluxes  due  to  the  series  and 
shunt  coils  pass  through  the  disk  and  develop  eddy  cur- 
rents therein,  which  react  on  the  fluxes  and  produce 
torque.  As  the  torque  is  dependent  upon  both  the  flux  of 
the  series  coil  and  that  of  the  shunt  coil,  it  is  proportional 
to  the  energy  which  is  to  be  measured.  To  render  the 
angular  velocity  of  the  disk  proportional  to  the  torque,  a 
permanent  brake  magnet  is  employed,  and  it  is  so  mounted 
as  to  allow  the  disk  to  revolve  between  its  poles.  The 
permanent  magnet  may  be  moved  radially  with  respect  to 
the  disk,  and  its  position  is  adjusted  to  obtain  the  proper 
retarding  force. 

It  is  necessary  to  have  the  series  and  shunt  fluxes  in 
time  quadrature  on  non-inductive  load  in  order  that  the 


248 


ALTERNATING-CURRENT   MACHINES. 


wattmeter  may  indicate  correctly  on  inductive  load.  To 
accomplish  this,  a  copper  band  with  a  small  gap  in  it, 
called  a  shading  coil,  is  placed  around  the  limb  of  the 
laminated  iron  core.  This  gap  is  closed  by  means  of  a 
resistance  wire  of  such  length  and  size  that  the  E.M.F. 


Fig.  182. 


Fig.  183. 


induced  in  this  band  by  the  alternating  shunt  flux  will 
send  a  current  through  it  of  such  value  that,  when 
combined  vectorially  with  the  current  in  the  shunt  coil, 
a  flux  at  right  angles  to  the  flux  in  the  series  coil  will 
result.  This  is  shown  in  Fig.  183,  where  <£  represents 
the  angle  by  which  the  current  in  the  series  coil  lags 
behind  the  impressed  E.M.F.  The  vectors  Esc  and  Isc 
represent  respectively  the  electromotive  force  and  the 
current  in  the  shading  coil.  The  resistance  of  the  shading 
coil  must  be  decreased  when  using  the  meter  on  circuits 
of  lower  frequency. 

Series  transformers  are  used  with  induction  wattmeters 
of  more  than  50  amperes  capacity,  and  potential  trans- 
formers are  employed  where  the  pressure  exceeds  300 


MOTORS.  249 

volts.  Polyphase  induction  wattmeters  consist  of  separate 
single-phase  elements  assembled  in  the  same  case.  The 
disks  are  mounted  on  a  common  shaft,  and  each  revolves 
in  its  own  field. 

SYNCHRONOUS   MOTORS. 

93.  Synchronous  Motors.  —  Any  excited  single-phase 
or  polyphase  alternator,  if  brought  up  to  speed,  and  if  con- 
nected with  a  source  of  alternating  E.M.F.  of  the  same 
frequency  and  approximately  the  same  pressure,  will  oper- 
ate as  a  motor.  The  speed  of  the  rotor  in  revolutions 
per  second  will  be  the  quotient  of  the  frequency  by  the 
number  of  pairs  of  poles.  This  is  called  the  synchronous 
speed;  and  the  rotor,  when  it  has  this  speed,  is  said  to  be 
running  in  synchronism.  This  exact  speed  will  be  main- 
tained throughout  wide  ranges  of  load  upon  the  motor  up 
to  several  times  full-load  capacity. 

To  understand  the  action  of  the  synchronous  motor, 
suppose  it  to  be  supplied  with  current  from  a  single 
generator.  The  following  discussion  refers  to  a  single- 
phase  motor,  but  may  equally  well  be  applied  to  the 
polyphase  synchronous  motor.  In  the  latter  case  each 
phase  is  to  be  considered  as  a  single-phase  circuit. 

Let  £x  =  E.M.F.  of  the  generator, 

E2  =  E.M.F.  of  the  motor  at  the  time  of  connec- 
tion with  the  generator, 
0  =  Phase  angle  between  El  and  E2, 
R  =  Resistance  of  generator  armature,  plus  that  of 
the    connecting   wires  and  of  the  motor 
armature,  and 
cuL  =  Reactance  of  the  above. 


250          ALTERNATING-CURRENT   MACHINES. 

The  resultant  E.M.F.,  E,  which  is  operative  in  sending 
current  through  the  complete  circuit,  is  found  by  combin- 

ing Ej  and  E2  with  each 
other  at  a  phase  differ- 
ence 0,  as  in  Fig.  184. 

Representing  the  angle 
between   E1   and   E   and 
E2  and   E  by    a    and    /? 
respectively,  it  follows  that 
E  =  E!  cos  a  +  E2  cos  /?. 

This  resulting  E.M.F.  sends  through  the  circuit  a  current 
whose  value  is 


and  it  lags  behind  E  by  an  angle  ^>,  such  that  tan  <£  =  -- 

R 

The  power  Pl  which  the  generator  gives  to  the  circuit  is 

P1  -  JBj/  cos  (a  -  0) 

and  the  power  P2  which  the  motor  gives  to  the  circuit  is 
P,  =  EJ  cos  09  +  $). 

Now,  if  in  either  of  the  above  expressions  for  power,  the 
cosine  has  any  other  •  value  than  unity,  then  the  power 
will  consist  of  energy  pulsations,  there  being  four  pulsa- 
tions per  cycle.  The  energy  is  alternately  given  to  and 
received  from  the  circuit  by  the  machine.  If  the  cosine 
be  positive,  the  amount  of  energy  in  one  pulsation,  which 
is  given  to  the  circuit,  will  exceed  the  amount  in  one 
of  the  received  pulsations.  The  machine  is  then  acting 
as  a  generator.  If  the  cosine  be  negative  the  opposite 
takes  place,  and  the  machine  operates  as  a  motor.  As  a 


MOTORS.  251 

and  /?  are  but  functions  of  Ev  E2,  and  6,  and  as  these  latter 
are  the  quantities  to  be  considered  in  operation,  it  is  desir- 
able to  eliminate  the  former.  From  the  foregoing 

P«- 

or 

E ?  cos  a  +  E,E7  cos  8  r 
Pl  =  — * —       — ^— L  2 [cos  a  cos  </>  +  sin  a  sin  <p\. 

Expanding,  this  becomes 

F 2 
P1  =  -^~  (cos2  a  cos  (j> '+  sin  a  cos  a  sin  </>) 

£  £ 
H ^r-2-  (cos  a  cos  /?  cos  0  +  sin  a  cos  /?  sin  0). 

But  2  x  —  #  =  «  +  /?; 

hence  cos  a  cos  /?  =  cos  0  +  sin  a  sin  /?, 

and  sin  a  cos  /?  =  —  sin  6  —  cos  a  sin  /?; 

also  cos2  a  =  i  —  sin2  a. 

Therefore 

Pj  =  -^-  (cos  (/>  —  sin2  a:  cos  <p  +  sin  a:  cos  a  sin  <£) 

£  E 
+  — t-2  (cos  6  cos  (56  +  sin  a  sin  y?  cos  0 

—  sin  6  sin  <£  —cos  a  sin  /?  sin  ^), 


+  -£  [E2  (sin  a  sin  /?  cos  0  —  cos  a  sin  /?  sin  < 
—  £x  (sin2  a  cos  $  —  sin  a  cos  a  sin  0)]. 


252  ALTERNATING-CURRENT   MACHINES. 

But,  since 

El  _  sin  /? 
E2       sin  a ' 

the  .second  term  reduces  to  zero,  and  therefore 
P!  =         f  ^TT"  cos  (0  +  #)  +  -          £l 

Similarly,  the  power  supplied  to  the  circuit  by  the  motor  is 

£,2 


If  there  were  no  losses  due  to  resistance,  etc.,  Pl  would  be 
numerically  exactly  equal  to  P2.  Neglecting  any  losses 
in  the  machines,  except  that  due  to  resistance,  the  alge- 
braic sum  of  P1  and  P2  is  equal  to  RP.  In  order  to 
determine  the  behavior  of  a  synchronous  motor  when  on 
a  given  circuit,  use  is  made  of  the  above  formula  for  power, 
and  each  case  must  be  considered  by  itself.  The  method 
of  procedure  is  shown  in  the  next  article. 

94.  Special  Case.  —  Suppose  a  single-phase  synchronous 
motor,  excited  so  as  to  generate  2100  volts,  to  be  con- 
nected to  a  generator  giving  2200  volts,  the  total  resist- 
ance of  the  circuit  being  2  ohms  and  the  reactance  i  ohm. 
Then  the  angle  (/>  of  current  lag  behind  the  resultant 

E.M.F.  has' a  value  tan  cf>  =  —  =  0.5,  whence  <£  =  26°  34'. 
K 

Calculations  of  Pl  and  P2  for  values  of  6  between  o°  and 
360°  have  been  made  using  the  formulae  of  the  preceding 
article,  the  results  being  embodied  in  the  form  of  curves 
in  Fig.  185.  Phase  differences,  6,  are  represented  as 
abscissae  and  Pl  and  P2,  in  kilowatts,  are  represented  as 


MOTORS. 


253 


ordinates.  An  enlargement  of  the  lower  portion  of  Fig.  185 
is  shown  in  Fig.  186.  The  ratio  of  P2  to  Pv  when  the 
former  is  negative  and  the  latter  positive,  and  when  all 
losses  excepting  the  copper  losses  are  neglected,  is  the 
motor  efficiency.  From  an  inspection  of  these  curves,  and 


4000 
3600 


3£  2000 


\ 


\ 


7 


7 


7 


7 


00         60         90        120        150       180       210       210       £70       300       330        360 
6    DEGREES 

Fig.  185. 

a  consideration  of  the  equations  from  which  the  curves  are 
derived,  the  following  conclusions  may  be  drawn :  — 

(a)  The   motor  will   operate   as   such  for  values   of  0 
between    175°   and    238°.     The   difference   between   these 
angles  may  be  termed  the  operative  range. 

(b)  The  generator  would  operate  as  a  motor  for  values 


254          ALTERNATING-CURRENT   MACHINES. 

of  0  between  133°  and  174°,  providing  the  motor  were 
mechanically  driven  so  as  to  supply  the  current  and  power; 
i.e.,  what  was  previously  the  motor  must  now  operate  as 
a  generator. 

(c)  The  motor,  within  its  operative  range,  can  absorb 
any  amount  of  power  between  zero  and  a  certain  maxi- 
mum. To  vary  the  amount  of  received  power,  the  motor 


Fig.  186. 

has  but  to  slightly  shift  the  phase  of  its  E.M.F.  in  respect 
to  the  impressed  E.M.F. ,  and  then  to  resume  running  in 
synchronism.  The  sudden  shift  of  phase  under  change 
of  load  is  the  fundamental  means  of  power  adjustment  in 
the  synchronous  motor.  It  corresponds  to  change  of  slip 
in  the  induction  motor,  to  change  of  speed  in  the  shunt 
motor,  and  to  change  of  magnetomotive  force  in  the 
transformer. 

(d)  For  all  values  of  the  received  power,  except  the 
maximum,  there  are  two  values  of  phase  difference  0. 
At  one  of  these  phase  differences  more  current  is  required 


MOTORS.  255 

for  the  same  power  than  at  the  other.     The  value  of  the 
current  in  either  case  can  be  calculated  as  follows :  — 

Since 


The  values  of  /  are  plotted  in  the  diagram.     The  efficiency 

p 
of  transmission  ?  =  — *-  is  also  different  for  the  two  values 

*i 
of  d.  It  is  also  represented  by  a  curve. 

If  the  phase  alteration,  produced  by  an  added  mechan- 
ical load  on  the  motor,  results  in  an  increase  of  power 
received  by  the  motor,  the  running  is  said  to  be  stable.  If, 
on  the  other  hand,  the  increase  of  load  produces  a  decrease 
of  absorbed  power,  the  running  is  unstable. 

(e)  If  for  any  reason  the  phase  difference  d,  between 
the  E.M.F.'s  of  the  motor  and  generator,  be  changed  to  a 
value  without  the  operative  range  for  the  motor,  the  motor 
will  cease  to  receive  as  much  energy  from  the  circuit  as  it 
gives  back,  and  it  will,  therefore,  fall  out  of  step.  Among 
the  causes  which  may  produce  this  result  are  sudden 
variations  in  the  frequency  of  the  generator,  variations 
in  the  angular  velocity  of  the  generator,  or  excessive 
mechanical  load  applied  to  the  motor.  In  slowing  down, 
all  possible  values  of  6  will  be  successively  assumed;  and 
it  may  happen  that  the  motor  armature  may  receive  suffi- 
cient energy  at  some  value  of  6  to  check  its  fall  in  speed, 
and  restore  it  to  synchronism,  or  it  may  come  to  a  stand- 
still. 

(/)  Under  varying  loads  the  inertia  of  the  motor  arma- 
ture plays  an  important  part.  The  shifting  from  one 
value  of  0  to  another,  which  corresponds  to  a  new  mechan- 


256          ALTERNATING-CURRENT   MACHINES. 

ical  load,  does  not  take  place  instantly.  The  new  value  is 
overreached,  and  there  is  an  oscillation  on  both  sides  of 
its  mean  value.  This  oscillation  about  the  synchronous 
speed  is  termed  hunting.  If  the  armature  required  no 
energy  to  accelerate  or  retard  it,  this  would  not  take 
place. 

(g)  The  maximum  negative  value  of  P2  —  that  is,  the 
maximum  load  that  the  motor  can  carry  —  is  evidently 
when  cos  (0  —  </>)=  —  i  or  when  0  —  (£>  =  180°.  The 
formula  for  the  power  absorbed  by  the  motor  then  reduces 


to 


(h)  The  operative  range  of  the  motor  can  be  determined 
by  making  P2  equal  to  zero.  By  transformation  the  for- 

mula then  becomes 

•p 

COS   (0   —  (/>)=    --  *-  COS  <j). 

El 

Two  values  of  (6  —  (j>)  result,  one  on  each  side  of  180°. 
In  the  case  under  consideration  cos  (&  —  </>)=  —.8537,  and 
0  -  (f)  =  211°  23'  or  148°  37'.  Since  <£  =  26°  34',  0  =  237° 
57'  or  175°  n'. 

95.   The  Motor  E.M.F.  —  To  determine  what  value  of 
E2  will  give  the  maximum  value  of  power  to  be  absorbed 
by  a  motor,  consider  E2  as  a  variable  in  the  equation  given 
in  (g)  above. 
Differentiating 

dP2m  _  2  £2  cos  <f>  -  E, 
dE2  ~'    V  R2  +  u?L2 

and  setting  this  equal  to  zero  and  solving, 
•p 

E2  =  -  —  -  =  1230  volts. 

2  COS0 


MOTORS. 


257 


At  this  voltage  the  maximum  possible  intake  of  the  motor 
is  605  K.W.  If  the  voltage  of  the  motor  be  above  this  or 
below  it,  its  maximum  intake  will  be  smaller. 

Remembering  that  the  current  lags  behind  the  resultant 
pressure  of  the  generator  and  motor  pressures  by  an  angle 
0,  which  is  solely  dependent  upon  co, 
L,  and  R,  it  will  be  easily  seen,  from 
an  inspection  of  Figs.  187,  188,  and 
189,  that  the  current  may  be  made 
to  lag  behind,  lead,  or  be  in  phase 
with  Ev  by  simply  altering  the  value 
of  E2.  This  may  be  done  by  vary- 
ing the  motor's  field  excitation.  A 
proper  excitation  can  produce  a  unit 
power  factor  in  the  transmitting 
line.  The  over-excited  synchronous 
motor,  therefore,  acts  like  a  con- 
denser in  producing  a  leading  cur- 
rent, and  can  be  made  to  neutralize  the  effect  of  induc- 
tance. The  current  which  is  consumed  by  the  motor  for 
a  given  load  accordingly  varies  with  the  excitation.  The 
relations  between  motor  voltage  and  absorbed  current  for 
various  loads  are  shown  in  Fig.  190. 

Synchronous  motors  are  sometimes  used  for  the  purpose 
of  regulating  the  phase  relations  of  transmission  lines. 
The  excitation  is  varied  to  suit  the  conditions,  and  the 
motor  is  run  without  load.  Under  such  circumstances  the 
machines  are  termed  synchronous  compensators. 

The  capacity  of  a  synchronous  motor  is  limited  by  its 
heating.  If  it  is  made  to  take  a  leading  current  in  order 
to  adjust  the  phase  of  a  line  current,  it  cannot  carry  its 
full  motor  load  in  addition  without  excessive  heating. 


CURRENT  IN  PHASE  WITH  E, 

Fig.  189. 


258 


ALTERNATING-CURRENT    MACHINES. 


MOTOR  VOLTAGE 


Fig.  190. 

96.  Starting  Synchronous  Motors.  —  Synchronous  motors 
do  not  have  sufficient  torque  at  starting  to  satisfactorily  come 
up  to  speed  under  load.  They  are,  therefore,  preferably 
brought  up  to  synchronous  speed  by  some  auxiliary  source 
of  power.  In  the  case  of  polyphase  systems  an  induction 
motor  is  very  satisfactory.  Its  capacity  need  be  but  yV  that 
of  the  large  motor.  Fig.  191  shows  a  75O-K.W.  quarter- 
phase  General  Electric  motor  with  a  small  induction  motor 
geared  to  the  shaft  for  this  purpose.  This  motor  may  be 
mechanically  disconnected  after  synchronism  is  reached. 
Before  connection  of  the  synchronous  motor  to  the  mains 
it  is  necessary  that  the  motor  should  not  only  be  in  syn- 
chronism, but  should  have  its  electromotive  force  at  a 
difference  of  phase  of  about  180°  with  the  impressed 
pressure.  To  determine  both  these  points  a  simple  device, 
known  as  a  synchronizer,  is  employed.  The  simplest  of 
these  is  the  connection  of  incandescent  lamps  across  a 
switch  in  the  circuit  of  the  generator  and  motor,  as  shown 
in  Fig.  192.  When  the  phase  difference  between  the  gen- 
erator and  motor  E.M.FSs  is  zero,  the  lamps  will  be 


MOTORS. 


259 


brightest,  and  when  the  phase  difference  is  180°,  the  lamps 
will  be  dark.  As  the  motor  comes  up  to  synchronous 
speed,  the  lamps  become  alternately  bright  and  dark.  As 
synchronism  is  approached,  these  alternations  grow  slower 
and  finally  become  so  slow  as  to  permit  closing  of  the 
main  switch  at  an  instant  when  the  lamps  are  dark. 


Fig.  IQI. 

Instead  of  connecting  the  synchronizing  device  directly 
in  the  main  circuit,  it  may  be  connected  in  series  with  the 
secondaries  of  two  transformers,  whose  primaries  are  con- 
nected respectively  across  the  generator  and  motor  ter- 
minals, as  shown  in  Fig.  193.  With  this  arrangement, 
maximum  brightness  of  the  lamps  may  indicate  that  the 
generator  and  motor  E.M.F.'s  are  either  in  phase  or  in 


260 


ALTERNATING-CURRENT   MACHINES. 


opposition,  according  to  the  manner  in  which  the  trans- 
former connections  are  made. 

Another  synchronizing  device  which  is  now  extensively 
used   is   known   as   the   synchroscope,   and   is   shown    in 


Fig.  192. 


Fig.  193- 

Fig.  194.  The  instrument  is  provided  with  a  pointer  which 
rotates  at  a  speed  proportional  to  the  difference  of  the 
generator  and  motor  frequencies,  the  direction  of  rotation 
showing  which  is  the  greater.  Thus,  if  the  motor  fre- 
quency is  too  high,  the  pointer  will  rotate  anti-clockwise. 
When  the  frequencies  are  identical,  the  pointer  assumes 
some  position  on  the  scale,  and  when  this  position  coin- 
cides with  the  index  at  the  top  of  the  scale,  the  main 


MOTORS.  26l 

switch  may  be  closed,  thus  connecting  the  two  machines 
together. 

Synchronous  motors  may  be  brought  up  to  speed  with- 
out any  auxiliary  source  of  power.  The  field  circuits  are 
left  open,  and  the  armature  is  connected  either  to  the  full 
pressure  of  the  supply,  or  to  this  pressure  reduced  by 
means  of  a  starting  compensator,  such  as  was  described 
in  §76.  The  magnetizing  effect  of  the  armature  ampere 


Fig.  194. 

turns  sets  up  a  flux  in  the  poles  sufficient  to  supply  a  small 
starting  torque.  When  synchronism  is  nearly  attained,  the 
fields  may  be  excited  and  the  motor  will  come  into  step. 
The  load  is  afterwards  applied  to  the  motor  through 
friction  clutches  or  other  devices.  There  is  great  danger  of 
perforating  the  insulation  of  the  field  coils  when  starting 
in  this  manner.  This  is  because  of  the  high  voltage 
produced  in  them  by  the  varying  flux.  In  such  cases 


262  ALTERNATING-CURRENT    MACHINES. 

each  field  spool  is  customarily  open-circuited  on  starting. 
Switches  which  are  designed  to  accomplish  this  purpose  are 
called  break-up  switches. 

97.  Parallel  Running  of  Alternators.  —  Any  two  alter- 
nators adjusted  to  have  the  same  E.M.F.   and  the  same 
frequency    may    be    synchronized    and    run    in    parallel. 
Machines  of  low  armature  reactance  have  large  synchro- 
nizing power,  but  may  give  rise  to  heavy  cross  currents,  if 
thrown  out  of  step  by  accident.     The  contrary  is  true  of 
machines   having   large    armature   reactance.     Gross    cur- 
rents due  to  differences  of  wave-form  are  also  reduced  by 
large  armature  reactance.    The  electrical  load  is  distributed 
between  the  two  machines  according  to  the  power  which  is 
being  furnished   by  the   prime   movers.     This  is   accom- 
plished, as  in  the  case  of  the  synchronous  motor,  by  a 
slight   shift  of  phase  between   the  E.M.F.'s  of  the   two 
machines.     The  difficulties  which  have  been  experienced  in 
the  parallel  running  of  alternators  have  almost  invariably 
been   due  to   bad   regulation  of  the  speed  of  the   prime 
mover.     Trouble  may  arise  from  the  electrical  side  if  the 
alternators  are  designed  with  a  large  number  of  poles. 
Composite  wound  alternators  should  have  their  series  com- 
pounding coils  connected  to  equalizing  bus  bars,  the  same 
as  compound  wound  direct-current  generators. 

SINGLE-PHASE  COMMUTATOR  MOTORS. 

98.  Single-Phase  Commutator  Motors.  —  If  the  current 
in  both  field  winding  and  armature  of  any  direct-current 
motor  be  periodically  reversed,  the  direction  of  rotation  of 
the  armature  will  remain  unchanged.     Therefore  direct- 
current  motors  might  be  operated  on  alternating-current 


MOTORS.  263 

circuits.  Shunt  motors  cannot  be  operated  satisfactorily 
when  fed  with  alternating  current,  because  the  reversals 
of  current  do  not  take  place  simultaneously  in  armature 
and  field  windings  owing  to  the  high  inductance  of  the 
latter  winding.  This  would  cause  momentary  currents  in 
the  armature  in  a  reversed  direction  and  would  tend  to 
produce  a  counter-torque,  thus  considerably  decreasing  the 
effective  torque. 

When    direct- current    series    motors    are    supplied    with 
alternating    current,    the    instantaneous    current    value    is 


Fig-  195. 

necessarily  the  same  in  both  armature  and  field  winding, 
and  therefore  no  counter-torque  is  developed.  The  direct- 
current  series  motor  with  various  modifications  may  be 
operated  on  alternating-current  circuits,  and  when  so  used 
is  termed  the  single-phase  series  motor,  or  the  single-phase 
commutator  motor.  It  is  essential  that  the  entire  magnetic 
circuits  of  motors  of  this  type  be  laminated  in  order  to 
decrease  the  otherwise  excessive  hysteresis  and  eddy  cur- 
rent losses.  Series  motors,  when  operated  on  alternating 
current,  produce  a  pulsating  torque  varying  from  zero  to  a 
certain  maximum  value. 

The  armature  of  a  single- phase  series  motor  is  similar 
to  that  of  the  direct-current  motor.  The  armature  of  a 
150  horse-power  single- phase  alternating-current  railway 
motor  is  represented  in  Fig.  195. 


264 


ALTERNATING-CURRENT    MACHINES. 


99.  Plain  Series  Motor.  —  Consider  a  direct-current 
armature  mounted  within  a  single-phase  alternating  mag- 
netic field,  as  in  Fig.  196.  When  the  armature  is  station- 
ary an  electromotive  force  will  be  induced  in  the  armature 
turns,  due  to  the  alternating  flux  which  passes  between  the 


Fig.  196. 

field  poles.  The  greatest  E.M.F.'s  will  be  induced  in  the 
turns  perpendicular  to  the  field  axis,  since  these  turns 
link  with  the  greatest  number  of  lines  of  force;  and  no 
E.M.F.'s  will  be  induced  in  the  turns  in  line  with  the 
field  axis.  The  directions  of  the  E.M.F.'s  induced  in  the 
armature  turns  by  the  change  in  field  flux  are  indicated  in 
the  figure  by  the  full  arrows,  and  it  is  seen  that  the  maxi- 
mum value  of  this  E.M.F.  is  across  EC.  As  in  trans- 
formers, the  effective  value  of  this  electromotive  force 
(§  59)  is 

,-,  2  7tf<&mN 

\/2    I08 


MOTORS.  265 

where  <bm  is  the  maximum  value  of  the  flux  entering  the 
armature  and  N  is  the  equivalent  number  of  armature 
turns. 

The  maximum  number  of  lines  of  force  linked  with  a 
single  turn  depends  upon  the  position  of  this  turn  in  the 
magnetic  field,  and  is  proportional  to  the  greatest  value  of 
<J>m  times  the  cosine  of  the  angle  of  displacement  of  the 
turn  from  the  position  AD.  Assuming  the  turns  to  be 
evenly  distributed  over  the  periphery  of  the  armature,  the 
average  value  of  the  maximum  flux  linked  with  the  arma- 
ture turns  will  be  -*  $m.  If  there  are  Na  conductors  on 

7T 

the   armature,    the    number   of   turns    connected    in    con- 

N 

tinuous   series    will    be    — ?  •     The   electromotive    forces 
2 

induced  in  the  upper  and  lower  groups  of  armature  turns 
are  added  in  parallel,  consequently  the  effective  number  of 

i       N        N 

turns  in  series  is  —  -    — —  =  — -  •     Therefore  the   equiva- 
22  4 

lent  number  of  armature  turns  may  be  expressed  as 

#-£.'&->.  (a) 

7T  4  2  7T 

Substituting  this  value  of  N  in  equation  (i),  the  E.M.F. 
induced  in  the  armature  winding  by  the  change  in  value  of 
the  field  flux  is 

ET  =  l^L,  (3) 

V  2    I08 

and  it  lags  90°  behind  field  flux  in  time. 

If  the  brushes  of  the  motor,  A  and  D,  are  placed  at  the 
points  shown  in  Fig.  196,  this  electromotive  force  will  not 


266  ALTERNATING-CURRENT   MACHINES. 

manifest  itself  externally,  since  it  consists  of  two  equal 
and  opposite  components  directed  toward  these  brushes. 
This  E.M.F.  appears,  however,  in  the  coils  short-circuited 
by  the  brushes,  as  will  be  shown  later.  The  current, 
which  enters  the  armature  by  way  of  the  brush  and  which 
traverses  the  two  halves  of  its  windings  in  parallel,  pro- 
duces an  armature  flux  of  maximum  value  3>am.  This  sets 
up  a  reactance  E.M.F.  in  the  armature  which  in  the  case 
of  uniform  gap  reluctance  can  be  similarly  expressed  as 

Ea=f-  (4) 


This  lags  90°  behind  the  current. 

When  the  armature  revolves,  there  are,  in  addition, 
electromotive  forces  induced  in  the  armature  conductors  as 
a  result  of  their  cutting  the  field  flux.  The  directions  of 
these  E.M.F. ,'s  are  indicated  by  the  dotted  arrows,  and  it 
is  seen  that  these  E.M.F. 's,  generated  by  the  rotation  of 
the  armature,  add  to  each  other  and  appear  on  the  com- 
mutator as  a  maximum  across  AD. 

The  average  value  of  the  electromotive  force  due  to  the 
rotation  of  the  armature  is 

Erotav  =  <bfNa  —  ID"  8,  §  42 

60 

where  V  is  the  armature  speed  in  rev.  per  min.  in  a 
bipolar  field,  and  <!>/  is  the  field  flux;  and  the  effective 
value  of  this  E.M.F.  is 

f'  V 

V2  io8       °° 

and  is  in  time  phase  with  the  field  flux,  but  appears  as  a 
counter  E.M.F.  at  the  brushes  AD. 


MOTORS. 


267 


When  an  alternating  current  is  passed  through  the  field 
coils,  the  alternating  field  flux  is  set  up,  and  this  flux  pro- 
duces a  reactive  E.M.F.  in  the  field  winding  lagging  90° 
behind  the  flux  in  phase,  exactly  as  in  a  choke  coil.  The 
magnitude  of  this  E.M.F.  is 


(6) 


where  3>/OT  is  the  maximum  value  of  the  field  flux,  and  Nf 
is  the  number  of  field  turns. 

The  electromotive  force,  E,  which  is  impressed  upon  the 
motor  terminals,  is.  equal  and  opposite  to  the  vectorial 


Fig.  197. 

sum  of  Ea,  Erot,  Ef,  and  the  IR  drop  of  the  armature 
and  field  windings,  as  shown  in  Fig.  197,  where  /  is  the 
current  flowing  through  the  field  and  armature,  and  4> 
represents  the  phase  of  the  flux.  In  this  diagram,  eddy 
current  and  hysteresis  losses  are  ignored.  The  impressed 
electromotive  force  is  therefore 

E  =  V(Erot  +  IR)2  +  (Ea  +  Ef)\  (7) 


268  ALTERNATING-CURRENT   MACHINES. 

100.  Characteristics  of  the  Plain  Series  Motor.  —  In  the 
series  motor,  the  same  current  passes  through  field  and 
armature  windings,  and,  if  uniform  reluctance  around  the 
air  gap  be  assumed,  then  the  armature  and  field  fluxes  will 
be  proportional  to  the  equivalent  armature  turns  and  field 
turns  respectively.  Therefore 

*M:*fi.-N:N,-^r:N}.  (i) 

Representing  the  ratio  of  the  field  turns  to  the  effective 

N 
armature  turns  by  T,  then  <I>/m  =  r<f>am,  and  Nj  =  r  • — a. 

Therefore    expressions     (4)     and     (6)     of     §  99     become 
respectively 


,  v       <bfmNa       , 

and  Ef  =  ~^—   •  fr. 

V2    I08 

Equation  (5 )  of  §  99  is 

<bf  AT  V 

•p       ^/ /»•  v  ft     .   _ 

•f^rot    ~  s       ? 

V  2   I08  °° 

which  then  reduces  to 

T          V  i          V 

Erot  =  —  Ea  —,  and   £ro<  =—£/—. 
/  oo  yr         oo 

Therefore  Ef  =  r2^. 

Neglecting   the   armature   and   field   resistance   drop,   the 
impressed  E.M.F.  becomes 


POWER  TRANSMISSION.  269 

which  is  the  fundamental  E.M.F.  equation  of  the  plain 
series  motor. 

The  power  factor  of  the  motor  is 

E^  rV  ,  , 

cos<£=  —  = =  ,         (3) 


and  the  current  supplied  to  the  motor  is 
E  E 


still  neglecting  the  motor  resistance. 

When  V=  6o/,  the  motor  is  said  to  run  at  synchronous 
speed  (bipolar  field).      The  power  factor  of  a  plain  series 

motor,  having  r  =  i,  when  running  at  this  speed,  is  — -, 

^5 

or  0.446,  and  for  values  of  T  other  than  unity  the  power 
factor  is  less  than  0.446.  It  is  true  that  if  the  resistance 
of  the  motor  be  considered,  the  power  factor  will  exceed 

this  value,  but  nevertheless  it  remains  extremely  low. 

•p 

The  current  intake  under  these  same  conditions  is  — — — •  • 

Vs*. 

When    the    motor   is  at  standstill,   V=  o,  and  the  power 

•p 

factor  is  zero.      The  current  intake  at  standstill  is 

2  Xa 

Hence  the  ratio  of  the  current  at  synchronism  to  the  cur- 
rent at  standstill  is  — =  -f-  -  =  0.894.  The  ratio  of  the 

VS         2 

torque  at  synchronous  speed  to  the  torque  at  standstill, 

since  it  varies   as  the   square  of  the  current,  is  ( — =)  -5- 

w' 


2/0          ALTERNATING-CURRENT   MACHINES. 

f—J  =  0.80,  which  shows  that  the  starting  torque  is  but 

little  greater  than  the  torque  at  synchronous  speed.  For 
railway  service  motors  are  required  having  large  starting 
torque  and  whose  torque  rapidly  decreases  as  the  speed 
of  the  motor  increases.  It  is  seen,  therefore,  that  inde- 
pendent of  its  low  power  factor,  the  plain  series  motor, 
having  uniform  magnetic  reluctance  around  the  air  gap, 
is  unsuitable  for  traction  and  for  similar  purposes. 

If,  however,  the  reluctance  of  the  air  gap  in  the  direction 
AD,  Fig.  196,  is  increased,  the  power  factor  and  speed- 
torque  characteristics  will  be  improved,  and  these  will 
depend  largely  upon  the  ratio  of  field  turns  to  effective 
armature  turns,  as  will  be  seen  by  considering  the  con- 
struction of  the  motor  to  be  such  that  the  proportion, 
equation  (i),  must  be  modified  by  the  introduction  of  a 
constant  considerably  greater  than  unity,  into  its  ante- 
cedents. A  motor  of  this  kind,  with  few  field  turns  com- 
pared to  armature  turns,  might  be  suitable  for  traction, 
but  more  important  improvements  have  been  made,  which 
will  now  be  discussed. 


loi.  Compensated  Series  Motors. — From  an  inspection  of 
Fig.  197  it  is  seen  that  the  power  factor  of  series  motors, may 
be  increased  by  increasing  IR  and  Erot,  or  by  decreasing  Ej 
and  Ea.  It  is  obvious  that  increasing  IR  signifies  an 
increase  in  losses,  thus  resulting  in  a  lower  efficiency. 
Erot  can  be  increased  by  increasing  the  number  of  armature 
turns.  Both  Ef  and  Ea  can  be  decreased  by  lowering  the 
frequency  without  affecting  Erot,  hence  low  frequencies 
are  desirable.  To  decrease  the  reactive  electromotive 
force  of  the  field,  it  is  necessary  that  the  reluctance 


MOTORS.  271 

of  the  magnetic  circuit  be  low,  i.e.,  small  air  gap  and 
low  flux  densities  in  the  iron,  in  order  that  the  required 
flux  can  be  produced  by  a  minimum  number  of  ampere 
turns.  The  armature  reactive  E.M.F.,  Ea,  is  not  essential 
to  the  operation  of  the  motor,  and  can  be  neutralized  by 
the  use  of  compensating  windings,  and  this  feature  of 
alternating- current  series  motors  is  a  very  important  one. 

The  compensating  winding  is  embedded  in  slots  in  the 
pole  faces,  as  shown  in  Fig.  198,  which  represents  a  West- 


Fig  198, 

inghouse  four-pole  compensated  single-phase  railway  motor 
with  its  armature  and  field  windings  removed.  The  num- 
ber of  turns  of  the  compensating  winding  is  adjusted  so 
as  to  set  up  a  magnetomotive  force  equal  and  opposite  to 
that  due  to  the  current  in  the  armature  coils.  The  com- 
pensating winding  may  be  energized  either  by  the  main 
current,  by  placing  this  winding  in  series  with  field  and 
armature,  or  by  an  induced  current,  which  is  obtained 
by  short-circuiting  the  compensating  winding  upon  itself, 
thus  utilizing  the  principle  of  the  transformer  in  that  the 
main  and  induced  currents  are  opposite  in  phase.  The 
former  method  of  neutralizing  Ea  is  known  as  conductive 
or  forced  compensation,  and  may  be  used  with  both  alter- 


2/2  ALTERNATING-CURRENT    MACHINES. 

nating  and  direct  currents,  and  the  latter  method  is  known 
as  inductive  compensation,  and  may  be  used  only  with  alter- 
nating current. 

Figs.  199  and  200  show  schematically  the  connections  of 


Fig.  199. 


Fig.  200. 


the  conductively  and  inductively  compensated  alternating- 
current    series    motors    respectively.      The    compensating 


winding  is   preferably  distributed   so   that  the   armature 
reactance  is  neutralized  as  completely  as   possible.     The 


MOTORS.  273 

current  flows  in  the  same  direction  in  all  of  the  con- 
ductors of  the  compensating  winding  embedded  in  one 
field  pole,  and  flows  in  the  opposite  direction  in  the  con- 
ductors embedded  in  the  adjacent  poles.  Fig.  201  illus- 
trates a  distributed  conductive-compensating  winding  of  a 
four-pole  machine. 

When  the  compensating  winding  completely  neutralizes 
the  armature  reactance,  the  impressed  electromotive  force 
(Eq.  7,  §99)  is 

E  -  V(Erot  +  IR)2  +  E/, 

where  R  is  the  resistance  of  the  motor  including  that  of  the 
compensating  winding.  If  the  resistance,  R,  be  neglected, 
then,  since 

y 

Erot  =  Ef,  §  100 

6o/r 
the  impressed  electromotive  force  becomes 


and  therefore  the  power  factor  is 

cos^  =  §*  =  -_ 

The  motor  current  is 


At  synchronous  speed  V  =  60 /,  and  therefore  the  power 

i 


factor  at  this  speed  becomes 


Vl    -f 


274  ALTERNATING-CURRENT  MACHINES. 

Still  neglecting  the  motor  resistance,  the  current  intake 

at  synchronous  speed  is  -  3  and  at  standstill  it 

X,Vi  +  ^ 

Tf 

is  •—  ?  consequently  the  ratio  of  the  current  at  synchronous 
xf 

speed   to   the    current   at    standstill  is    — —  .      Since 

Vi  +  T2 

torque  varies  as  the  square  of  the  current,   the  ratio   of 
the  torque  at  synchronous  speed  to  the  starting  torque  is 
'£ 
—  •      Hence  it  follows  that  the  speed-torque  charac- 

I   +  r2 

teristics  of  a  compensated  series  motor  may  be  adjusted 
to  the  required  conditions  by  properly  proportioning  the 
number  of  armature  and  field  turns. 

Compensated  series  motors  are  well  suited  for  traction. 
The  performance  curves  of  the  25o-horse-power  25-cycle 
Westinghouse  conductively  compensated  single-phase  series 
motor  used  on  the  New  York,  New  Haven  and  Hartford 
Railroad  are  shown  in  Fig.  202.  Each  locomotive  is 
equipped  with  four  of  these  motors  operating  at  225  volts, 
which  is  procured  by  step-down  transformation  from  over- 
head 1 1,000- volt  trolley.  On  parts  of  the  road  the  motors 
operate  on  direct  current,  the  current  being  supplied  directly 
to  the  motors. 


102.  Sparking  in  Series  Motors.  —  The  principal  diffi- 
culty encountered  in  the  operation  of  single-phase  series 
motors  is  the  sparking  at  the  brushes.  This  is  caused  by 
the  local  currents  produced  by  the  E.M.F.  generated  in 
the  armature  turns  short-circuited  by  the  brushes,  due  to 
the  periodic  reversals  of  the  field  flux.  With  the  brushes 
located  in  the  neutral  position  with  respect  to  the  E.M.F. 


POWER  TRANSMISSION, 


275 


of  rotation,  the  short-circuited  turns  are  perpendicular  to 
the  axis  of  the  field  flux,  and  therefore  the  flux  linked  with 
these  turns  is  a  maximum.  The  electromotive  force  gener- 
ated in  a  short-circuited  armature  section  is 


Es- 


where  Ns  is  the  equivalent  number  of  armature  turns  per 
section  which  is  short-circuited  by  a  brush.     If  the  resist- 


rflLES  PER  HOUR 


200       40 

30 

100      20 

10 


z 


200       100        600        £00       1000     1200      1400      1600 
AMPERES 

Fig.  202. 


ance  of  the  section  be  small,  an  enormous  current  will  flow, 
and  will  cause  excessive  heating  of  the  brush,  commutator 
segments,  and  armature  conductors. 
In  order  to   decrease   this   local   current,   the  E.M.F. 


276  ALTERNATING-CURRENT   MACHINES. 

induced  in  each  section  may  be  decreased  and  the  resist- 
ance thereof  may  be  increased.  From  an  inspection  of 
the  preceding  formula  it  is  seen  that  Es  may  be  decreased 
by  reducing  the  number  of  armature  turns  per  section,  by 
lowering  the  maximum  value  of  the  flux,  and  by  lowering 
the  frequency.  Thus  single-phase  series  motors  are 
usually  provided  with  more  than  two  poles  and  with 
many  commutator  segments,  and  are  designed  to  operate 
on  low  frequency  circuits.  A  simple  way  to  increase  the 


Fig.  203. 

resistance  of  the  armature  sections  involves  the  use  of 
preventive  or  resistance  leads,  which  are  connected  between 
armature  conductors  and  commutator  segments,  as  illus- 
trated in  Fig.  203.  It  has  been  shown  by  experiment  that 
the  losses  are  a  minimum  when  the  resistance  of  the  pre- 
ventive leads  is  so  proportioned  that  the  short-circuit  cur- 
rents and  normal  currents  are  equal.  The  resistance 
leads  are  usually  of  German  silver  and  have  a  large  current- 
carrying  capacity.  They  are  placed  in  the  same  slots  as 
the  armature  conductors,  and  usually  at  the  bottom  thereof. 
Only  a  few  of  these  leads  are  in  circuit  at  any  instant, 
and,  when  the.  armature  rotates,  all  of  the  preventive  leads 
carry  current  in  turn,  hence  the  average  loss  of  power, per 


MOTORS. 


277 


lead  is  small.  As  the  heating  effect  of  the  short-circuit 
current  depends  upon  the  duration  of  the  short  circuit,  it 
is  essential  that  the  brushes  be  made  quite  narrow. 

103.  Repulsion  Motors.  — The  repulsion  motor  con- 
sists of  a  field  resembling  the  stator  of  the  single-phase 
induction  motor,  and  an  armature  which  is  similar  to  the 
armatures  of  direct-current  and  alternating-current  series 
motors.  The  armature  winding  always  remains  short- 


Fig.  204. 

circuited  in  a  line  inclined  at  a  definite  angle  with  the 
field  axis,  this  being  accomplished  by  means  of  brushes, 
bearing  on  the  commutator,  which  are  joined  together 
by  a  conductor  of  low  resistance.  The  field  winding  is 
supplied  with  single-phase  alternating  current.  The  fact 
that  the  armature  and  field  windings  are  electrically  dis- 
tinct makes  it  possible  to  operate  the  motor  on  high  volt- 
age systems,  the  armature  winding  being  so  adjusted  that 
the  currents  therein  can  be  commutated  satisfactorily. 


278 


ALTERNATING-CURRENT    MACHINES. 


The  pulsating  flux  through  the  armature,  produced  by 
the  alternating  current  in  the  field  winding,  may  be  con- 
sidered as  the  resultant  of  two  components,  one  in  the 
direction  of  the  brush  axis,  and  the  other  perpendicular 
thereto;  these  being  represented  in  Fig.  204  respectively 
by  OA  and  AB.  The  component  OA  produces  an  E.M.F. 
in  the  armature  conductors  and  causes  a  current  to  flow 


Fig.  205. 

through  them.     The  other  component,  AB,  reacts  upon 
this  armature  current,  thereby  developing  torque. 

To  represent  the  action  of  a  repulsion  motor  more 
clearly,  the  field  winding  may  be  considered  as  composed 
of  two  parts  placed  at  right  angles  to  each  other,  as  at 
M  and  AT,  Fig.  205,  and  the  brushes  may  be  located  in 
line  with  one  of  them.  When  the  rotor  is  stationary,  the 
pulsating  flux  from  poles  MM  causes  the  flow  of  current  in 
the  short-circuited  armature,  the  effect  of  which  is  the  pro- 
duction of  a  flux  opposite  and  nearly  equal  to  that  which 
caused  the  current  flow.  The  flux  from  poles  MM  is 
thereby  reduced,  thus  resulting  in  increased  stator  current 


POWER  TRANSMISSION. 


279 


and  flux  from  poles  NN.  Neglecting  iron  losses,  this  flux 
will  be  in  phase  with  the  line  current,  whereas  the  phase  of 
the  armature  current  is  opposite  to  that  of  the  current  in 
coils  MM.  Hence  the  flux  from  poles  NN  is  in  phase 
with  the  armature  current,  and  their  product,  torque, 


Fig.  206. 

retains  its  sign  as  both  reverse  in  direction  together.  The 
repulsion  motor  exerts  its  maximum  torque  at  starting,  and 
this  torque  decreases  with  decreasing  current  and  with 
increased  speed,  and  consequently  this  type  of  commuta- 
tor motor  is  well  adapted  for  single-phase  traction.  The 
power  factor  of  repulsion  motors  is  low  at  starting  and 
rises  rapidly  as  the  speed  increases.  Repulsion  motors 
may  be  operated  on  25  ^  or  even  on  60  ~~  supply  circuits. 
The  fact  that  the  repulsion  motor  may  be  converted 


280 


ALTERNATING-CURRENT  MACHINES. 


readily  into  a  single-phase  induction  motor,  by  simply 
short-circuiting  the  entire  commutator,  thus  changing  the 
armature  to  a  squirrel-cage  rotor,  has  led  to  the  design  of 
single-phase  induction  motors  which  start  and  come  up 
to  speed  as  repulsion  motors.  A  motor  of  this  type,  manu- 


Fig.  207. 

factured  by  the  Wagner  Electric  Company,  is  shown  in 
Fig.  206.  Upon  reaching  normal  speed,  a  centrifugal 
device,  shown  in  the  figure,  causes  the  commutator  bars 
to  be  short-circuited,  and  the  brushes  are  simultaneously 
lifted  from  the  commutator.  The  results  of  tests  made 
upon  this  type  of  motor  are  represented  in  the  curves  of 
Fig.  207. 

104.  Series-Repulsion  Motor.  —  A  single-phase  railway 
motor  which  embodies  many  of  the  best  features  of  the 
repulsion  motor  and  of  the  compensated  series  motor,  and 
therefore  called  the  series-repulsion  motor,  has  recently 
been  developed  by  the  General  Electric  Company. 


POWER  TRANSMISSION. 


281 


The  windings  resemble  those  of  a  series  motor,  and  the 
armature  has  a  fractional-pitch  winding  such  as  is  used 
on  direct-current  motors.  The  connections  of  the  motor 
circuits  for  the  starting  and  running  positions  are  shown 
in  Fig.  208. 

The  motor  starts  as  a  repulsion  motor  and  possesses 
all  the  characteristics  thereof  when  in  this  position.     The 


Fig.2o8. 

compensating  coils  differ  from  those  on  series  motors  in 
that  they  have  twice  as  many  turns  as  there  are  on  the 
armature,  and  therefore  the  armature  current  at  starting 
is  twice  as  large  as  the  current  flowing  through  the  field 
and  compensating  windings.  Thus  the  starting  torque  is 
twice  as  great  as  would  be  obtained  for  the  same  current 
with  a  compensated  series  motor.  When  in  the  running 
position,  the  motor  characteristics  are  similar  to  those  of 
the  compensated  series  motor,  but  it  possesses  an  advantage 
over  the  latter  in  regard  to  sparking. 

At  starting  there  is  very  little  sparking  at  the  brushes, 


282 


ALTERNATING-CURRENT   MACHINES. 


but  this  increases  up  to  a  certain  value  of  voltage  induced 
in  the  short-circuited  armature  sections.  This  value 
practically  corresponds  to  that  which  gives  good  com- 
mutation in  the  running  position.  Better  inherent  com- 
mutation is  the  chief  advantage  of  the  series-repulsion 
motor. 

PROBLEMS. 

1.  Determine  approximately  the   full-load  efficiency  of  a   certain 
i5-horse-power  three-phase  six-pole  5o-cycle  induction   motor  which 
makes  950  revolutions  per  minute  when  carrying  full  load. 

2.  What  torque  does  the  motor  of  the   preceding  problem  exert 
when  operating  under  full  load  ? 

3.  Calculate  the  leakage  reactance  per  phase  of  a  5ooo-volt  three- 
phase  25-cycle  30-pole  induction  motor  having  a  three-phase  wound 


8TATOR 
1T82 


Fig.  209. 

rotor  (both  stator  and  rotor  have  Y-connected  full-pitch  windings). 
The  dimensions  of  the  slots  in  inches  are  indicated  in  Fig.  209,  and 
the  constants  of  the  motor  are: 

Rotor  diameter „ . .  „ 118        in. 

Total  number  of  primary  slots 450 

Total  number  of  secondary  slots 720 

Series  conductors  per  primary  slot , 8 

Series  conductors  per  secondary  slot J 

Length  of  slots 29.5     in. 

Length  of  end  connection  per  primary  turn 48.5     in. 

Depth  of  laminations  back  of  slots 2.36  in. 


PROBLEMS.  283 

4.  Calculate  the  exciting  current  per  phase  of  the  induction  motor 
of  the  preceding  problem. 

5.  The  stator  resistance  per  phase  of  the  induction  motor  of  prob- 
lem 3  is  1.7  ohms,  and  the  rotor  resistance  per  phase  is  .02  ohm, 
which,  when  reduced  to  the  stator  circuit,  is  2  ohms.     Determine  the 
motor  performance  curves  from  its  circle  diagram,  this  being  based 
upon  the  results  of  the  two  preceding  problems. 

6.  Plot  curves  of  Pl  and  P2  for  a  single-phase  synchronous  motor, 
excited  so  as  to  generate  1000  volts,  connected  to  a  single-phase  alter- 
nator having  an  E.M.F.  of  1200  volts,  the   total   resistance   of  the 
circuit  being  1.75  ohms  and  the  total  impedance  2  ohms. 


284 


ALTERNATING  CURRENT   MACHINES. 


CHAPTER   VIII. 

CONVERTERS. 

105.  The  Converter.  —  The  converter  is  a  machine  hav- 
ing one  field,  and  one  armature,  the  latter  being  supplied 
with  both 'a  direct-current  commutator  and  alternating- 
current  slip-rings.  When  brushes,  which  rub  upon  the 
slip-rings,  are  connected  with  a  source  of  alternating 
current  of  proper  voltage,  the  armature  will  rotate  syn- 
chronously, acting  the 
same  as  the  armature  of  a 
synchronous  motor.  While 
so  revolving,  direct  current 
can  be  taken  from  brushes 
rubbing  upon  the  commu- 
tator. The  intake  of  cur- 
rent from  the  alternating- 
current  mains  is  sufficient 
to  supply  the  direct-current 
circuit,  and  to  overcome 
the  losses  due  to  resistance, 
friction,  windage,  hyster-  Flg>  2I°- 

esi.s,  and  eddy  currents.  The  windings  of  a  converter 
armature  are  closed,  and  simply  those  of  a  direct-current 
dynamo  armature  with  properly  located  taps  leading  to  the 
slip-rings.  Each  ring  must  be  connected  to  the  armature 
winding  by  as  many  taps  as  there  are  pairs  of  poles  in 
the  field.  These  taps  are  equidistant  from  each  other. 


CONVERTERS. 


285 


There  may  be  any  number  of  rings  greater  than  one. 
A  converter  having  n  rings  is  called  an  ;/-ring  converter. 

The  taps  to  successive  rings  are  -th  of  the  distance  be- 

11 

tween  the  centers  of  two  successive  north  poles  from  each 
other.  Fig.  210  shows  the  points  of  tapping  for  a  3-ring 
multipolar  converter. 

A  converter  may  also  be  supplied  with  direct  current 


Fig.  211. 

through  its  commutator,  while  alternating  current  is  taken 
from  the  slip-rings.  Under  these  circumstances  the 
machine  is  termed  an  inverted  converter.  Converters  are 
much  used  in  lighting  and  in  power  plants,  sometimes 
receiving  alternating  current,  and  at  other  times  direct 
current.  In  large  city  distributing  systems  they  are  often 
used  in  connection  with  storage  batteries  to  charge  them 


286          ALTERNATING-CURRENT    MACHINES. 

from  alternating-current  mains  during  periods  of  light 
load,  and  to  give  back  the  energy  during  the  heavy  load. 
They  are  also  used  in  transforming  alternating  into  direct 
currents  for  electrolytic  purposes.  A  three-phase  machine 
for  this  purpose  is  shown  in  Fig.  211. 

A  converter  is  sometimes  called  a  rotary  converter  or 
simply  a  rotary. 

106.  E.M.F.  Relations.  —  In  order  to  determine  the  re- 
lations which  exist  between  the  pressures  available  at  the 
various  brushes  of  a  converter, 

Let  Ed  =  the  voltage  between  successive  direct-current 

brushes. 

En  =  the  effective  voltage  between  successive  rings 
of  an  «-ring  converter. 

a  =  the  maximum  E.M.F.  in  volts  generated  in  a 
single  armature  inductor.  This  will  exist 
when  the  conductor  is  under  the  center  of  a 
pole. 

b  =  the  number  of  armature  inductors  in  a  unit 
electrical  angle  of  the  periphery.  The 
electrical  angle  subtended  by  the  centers  of 
two  successive  poles  of  the  same  polarity 
is  considered  as  2?r 

The  E.M.F.  generated  in  a  conductor  may  be  considered 
as  varying  as  the  cosine  of  the  angle  of  its  position  relative 
to  a  point  directly  under  the  center  of  any  north  pole,  the 
angles  being  measured  in  electrical  degrees.  At  an  angle 
A  Fig.  212,  the  E.M.F.  generated  in  a  single  inductor  G 
is  a  cos  ft  volts.  In  an  element  df$  of  the  periphery  of 
the  armature  there  are  bdft  inductors,  each  with  this 
E.M.F.  If  connected  in  series  they  will  yield  an  E.M.F. 


CONVERTERS. 


287 


of  ab  cos  ft  d($  volts.     The  value  of  ab  can  be  determined 
if  an  expression  for  the  E.M.F.  between  two  successive 

direct-current  brushes  be 
determined  by  integration, 
and  be  set  equal  to  this 
value  Ed  as  follows  : 


1=  I        a< 

*J        IT 


=  2  ab. 


.£* 

2 


Fig.  212. 


successive  rings  is    — 
n 


In  an  #-ring  converter,  the 
electrical  angular  distance 
between  the  taps  for  two 

The  maximum  E.M.F.  will  be 


generated  in  the  coils  between  the  two  taps  for  the  succes- 
sive rings,  when  the  taps  are  at  an  equal  angular  distance 
from  the  center  of  a  pole,  one  on  each  side  of  it,  as  shown 
in  the  figure.  This  maximum  E.M.F.  is 


ab  cos  3d3  =  2  ab  sin  - 

n 


The  effective   voltage  between  the  successive  rings  is 
therefore 


By  substituting  numerical  values  in  this  formula,  it  is 
found  that  the  coefficient  by  which  the  voltage  between 


288          ALTERNATING-CURRENT    MACHINES. 

the  direct -current  brushes  must  be  multiplied  in  order  to 
get  the  effective  voltage  between  successive  rings  is  for 

2  rings 0.707 

3  rings 0.612 

4  rings 0.500 

6  rings  .     .  '£.,  >..   ....     ...     .  0.354 

In  practice  there  is  a  slight  variation  from  these  co-effi- 
cients due  to  the  fact  that  the  air-gap  flux  is  not  sinusoid- 
ally  distributed. 

107.  Current  Relations.—  In  the  following  discussion  it  is 
assumed  that  a  converter  has  its  field  excited  so  as  to 
cause  the  alternating  currents  in  the  armature  inductors  to 
lag  1 80°  behind  the  alternating  E.M.F.  generated  in  them. 

The  armature  coils  carry  currents  which  vary  cyclically 
with  the  same  frequency  as  that  of  the  alternating-current 
supply.  They  differ 
widely  in  wave-form  from 
sine  curves.  This  is  be- 
cause they  consist  of  two 
currents  superposed  upon 
each  other.  Consider  a 
coil  B,  Fig.  213.  It  car- 
ries a  direct  current  whose 

value  —  is  half  that  car- 

ried  by  one  direct-current       / 

brush,  and  it  reverses  its  ' 

direction  every  time  that  Fig* al3' 

the    coil    passes    under   a   brush.      The   coil,   as  well   as 

all  others  between  two  taps  for  successive  slip-rings,  also 

carries  an  alternating  current.     This  current  has  its  zero 


CONVERTERS. 


289 


value  when  the  point  A,  which  is  midway  between  the 
successive  taps,  passes  under  the  brush.  The  coil  being 
$  electrical  degrees  ahead  of  the  point  A,  the  alternating 

current  will  pass  through  zero  -^—   of  a  cycle  later  than 

2  71 

the  direct  current.  The  time  relations  of  the  two  currents 
are  shown  in  Fig.  214. 

To   determine   the  maximum  value  of  the   alternating 
current  consider  that,  after  subtracting  the  machine  losses, 


Fig.  214. 

the  alternating-current  power  intake  is  equal  to  the  direct- 
current  power  output.  Neglecting  these  losses  for  the 
present,  if  En  represents  the  pressure  and  In  the  effective 
alternating  current  in  the  armature  coils  between  the  suc- 
cessive slip-rings,  then  for  the  parts  of  the  armature  wind- 
ings covered  by  each  pair  of  poles 

Edld  =  nEJn 

Ed     .     x 
=  n  —— -  sin  -  /„. 

A/2  n 

Therefore,  the  maximum  value  of  the  alternating  current  is 


In    = 


The  time  variation  of  current  in  the  particular  coil  B  is 
obtained  by  taking  the  algebraic  sum  of  the  ordinates  of 


290 


ALTERNATING-CURRENT    MACHINES. 


the  two  curves.  This  yields  the  curve  shown  in  Fig.  215. 
Each  inductor  has  its  own  wave-shape  of  current, 
depending  upon  its  angular  distance  0  from  the  point  A. 
Converter  coils,  therefore,  alternately  functionate  as  motor 
and  as  generator  coils. 


Fig.  215. 

1 08.  Heating  of  the  Armature  Coils.  —  The  heating 
effect  in  an  armature  coil  due  to  a  current  of  such  peculiar 
wave-shape  as  that  shown  in  Fig.  215  can  be  determined 
either  graphically  or  analytically.  The  graphic  determina- 
tion requires  that  a  new  curve  be  plotted,  whose  ordinates 
shall  be  equal  to  the  squares  of  the  corresponding  current 
values.  The  area  contained  between  this  new  curve  and 
the  time  axis  is  then  determined  by  means  of  a  planimeter. 
The  area  of  one  lobe  is  proportional  to  the  heating  value 
of  the  current.  This  value  may  be  determined  for  each 
of  the  coils  between  two  successive  taps.  An  average  of 
these  values  will  give  the  average  heating  effect  of  the 
currents  in  all  the  armature  coils.  The  heating  is  different 
in  the  different  coils.  It  is  a  maximum  for  coils  at  the 
points  of  tap  to  the  slip-rings  and  is  a  minimum  for  coils 
midway  between  the  taps. 


CONVERTERS.  2QI 

109.  Capacity  of  a  Converter.  —  As  the  result  of  a 
rather  involved  analysis  it  is  found  that  a  machine  has 
different  capacities,  based  upon  the  same  temperature  rise, 
according  to  the  number  of  slip-rings,  as  shown  in  the  fol- 
lowing table.  The  armature  is  supposed  to  have  a  closed- 
coil  winding. 

CONVERTER   CAPACITIES. 
USED  AS  A  KILOWATT  CAPACITY 

Direct-current  generator       .-''.. 100 

Single-phase  converter 85 

Three-phase  converter 134 

Four-phase  converter 164 

Six-phase  converter 196 

Twelve-phase  converter „     .     .  227 

The  overload  capacity  of  a  converter  is  limited  by  com- 
mutator performance  and  not  by  heating.  As  there  is  but 
small  armature  reaction,  the  limit  is  much  higher  than  is 
the  case  with  a  direct-current  generator. 

no.  Starting  a  Converter.  —  Converters  may  be  started 
and  be  brought  up  to  synchronism  by  the  same  methods 
which  are  employed  in  the  case  of  synchronous  motors. 
It  is  preferable,  however,  that  they  be  started  from  the 
direct-current  side  by  .the  use  of  storage  batteries  or  other 
sources  of  direct  current.  They  may  be  brought  to  a  little 
above  synchronous  speed  by  means  of  a  starting  resistance 
as  in  the  case  of  a  direct-current  shunt  motor,  and  then, 
after  disconnecting  and  after  opening  the  field  circuit,  the 
connections  with  the  alternating-current  mains  may  be 
made.  This  will  bring  it  into  step. 

in.  Armature  Reaction.  —  The  converter  armature  cur- 
rents give  rise  to  reactions  which  consist  of  direct-current 


292 


ALTERNATING-CURRENT   MACHINES. 


generator  armature  reactions  superposed  upon  synchronous 
motor  armature  reactions.  It  proves  best  in  practice  to 
set  the  direct-current  brushes  so  as  to  commutate  the  cur- 
rent in  coils  when  they  are  midway  between  two  succes- 


Fig.  216. 

sive  poles.  The  direct-current  armature  reaction,  then,  con- 
sists in  a  cross-magnetization  which  tends  to  twist  the  field 
flux  in  the  direction  of  rotation.  When  the  alternating 
currents  are  in  phase  with  the  impressed  E.M.F.  they  also 
exert  a  cross-magnetizing  effect  which  tends  to  twist  the 


CONVERTERS.  293 

field  flux  in  the  opposite  direction.  The  result  of  this  neu- 
tralization is  a  fairly  constant  distribution  of  flux  at  all 
loads.  Within  limits  even  an  unbalanced  polyphase  con- 
verter operates  satisfactorily.  There  is  no  change  of  field 
excitation  necessary  with  changes  of  load. 

The  converter  is  subject  to  hunting  the  same  as  the 
synchronous  motor.  As  its  speed  oscillates  above  and 
below  synchronism,  the  phase  of  the  armature  current,  in 
reference  to  the  impressed  E.M.F.,  changes.  This  results 
in  a  distortion  of  the  field  flux,  of  varying  magnitude. 
This  hunting  is  much  reduced  by  placing  heavy  copper 
circuits  near  the  pole  horns  so  as  to  be  cut  by  the  oscillat- 
ing flux  from  the  two  horns  of  the  pole.  The  shifting  of 
flux  induces  heavy  currents  in  these  circuits  which  oppose 
the  shifting.  Fig.  216  shows  copper  bridges  placed  be- 
tween the  poles  of  a  converter  for  this  purpose. 

When  running  as  an  inverted  converter  from  a  direct- 
current  circuit,  anything  which  tends  to  cause  a  lag  of  the 
alternating  current  behind  its  E.M.F.  is  to  be  avoided. 
The  demagnetization  of  the  field  by  the  lagging  current 
causes  the  armature  to  race  the  same  as  in  the  case  of  an 
unloaded  shunt  motor  with  weakened  fields.  Converters 
have  been  raced  to  destruction  because  of  the  enormous 
lagging  currents  due  to  a  short  circuit  on  the  alternating- 
current  system. 

112.  Regulation  of  Converters The  field  current  of  a 

converter  is  generally  taken  from  the  direct-current 
brushes.  By  varying  this  current  the  power  factor  of  the 
alternating-current  system  may  be  changed.  This  may 
vary,  through  a  limited  range,  the  voltage  impressed 
between  the  slip-rings.  As  the  direct-current  voltage 


294 


ALTERNATING-CURRENT   MACHINES. 


Step- down 
Transformer. 


bears  to  the  latter  a  constant  ratio  it  may  also  be  varied. 

This  is,  however,  an  uneconomical  method  of  regulation. 

Converters  are  usually  fed  through  step-down  transform- 
ers. In  such  cases 
there  are  two  com- 
mon methods  of  regu- 
lation, which  vary  the 
voltage  supplied  to 
the  converter's  slip- 
of  Stillwell,  which  is 


Fig.  217. 

rings.      The   first 


is  the  method 
shown  in  the  diagram,  Fig.  217. 

The  regulator  consists  of  a  transformer  with  a  sectional 


Fig.  218. 


CONVERTERS.  295 

secondary.  Its  ratio  of  transformation  can  be  altered  by 
moving  a  contact-arm  over  blocks  connected  with  the 
various  sections,  as  shown  in  the  diagram.  The  primary 
of  the  regulator  is  connected  with  the  secondary  terminals 
of  the  step-down  transformer.  The  sections  of  the  second- 
ary, which  are  in  use,  are  connected  in  series  with  the  step- 
down  secondary  and  the  converter  windings. 

The  second  method  of  regulation  is  that  employed  by 
the  General  Electric  Co.  The  ratio  of  transformation  of 
a  regulating  transformer,  which  is  connected  in  circuit  in 
the  same  manner  as  the  Stillwell  regulator,  is  altered  by 
shifting  the  axes  of  the  primary  and  secondary  coils  in 
respect  to  each  other.  Fig.  2 1 8  shows  such  a  transformer, 
the  shifting  being  accomplished  by  means  of  a  small, 
direct-current  motor  mounted  upon  the  regulator.  The 
primary  windings  are  placed  in  slots  on  the  interior  of  a 
laminated  iron  frame,  which  has  the  appearance  of  the 
stator  of  an  induction  motor.  The  secondary  windings  are 
placed  in  what  corresponds  to  the  slots  of  the  rotor  core. 
The  winding  is  polar  ;  and  if  the  secondary  core  be  rotated 
by  an  angle  corresponding  to '  the  distance  between  two 
successive  poles,  the  action  of  the  regulator  will  change 
from  that  of  booster  to  that  of  crusher. 

Another  method  of  converter  regulation,  sometimes 
used  in  railway  work,  makes  use  of  reactance  coils,  con- 
nected between  the  step-down  transformer  coil  terminals 
and  the  slip-rings  of  the  converter,  as  well  as  of  an  ordi-' 
nary  series  compounding  coil  on  the  field-cores  of  the  con- 
verter. The  series  and  shunt  field  coils  are  so  adjusted 
that  the  converter  takes  a  lagging  current  at  no  load  and 
a  leading  current  at  full  load.  The  step-down  transformer 
voltage  being  assumed  as  constant,  the  voltage  impressed 


296  ALTERNATING-CURRENT   MACHINES. 

upon  the  slip-rings  will  be  the  remainder  resulting  from 
the  vector  subtraction  of  the  reactance  drop  from  the 
constant  voltage.  On  heavy  loads  and  leading  currents 
this  remainder  is  greater  than  the  constant  voltage.  There 
is  therefore  a  constantly  increasing  voltage  impressed 
upon  the  slip-rings  as  the  load  increases.  Too  large  a 
reactance,  however,  is  liable  to  introduce  pulsation  troubles. 

In  Europe  some  use  is  made  of  a  small  auxiliary  alter- 
nator mounted  upon  the  shaft  of  the  converter  and  oper- 
ating synchronously  with  it.  By  varying  and  reversing 
the  field  excitation  of  this  alternator,  whose  armature 
phases  are  connected  between  the  transformer  terminals 
and  the  slip-rings  of  the  converter,  it  may  be  caused  to 
act  as  a  booster  or  as  a  crusher. 

Recently  converters  have  been  constructed  in  a  manner 
that  permits  of  altering  their  ratios  of  voltage  conversion 
by  changing  the  distribution  of  flux  in  the  air  gap.  Non- 
sine  waves  of  E.M.F.  are  then  induced  in  the  armature 
inductors.  The  ratios  of  voltage  conversion  hitherto 
deduced  upon  the  assumption  of  sine  wave-forms  do  not 
then  hold.  The  change  of  flux  distribution  is  accomplished 
by  splitting  each  pole  into  sections  along  axial  planes. 
The  sections  are  then  subjected  to  different  magneto- 
motive forces  which  may  be  independently  varied  during 
operation. 

113.  Mercury  Vapor  Converter.  —  A  mercury  vapor 
converter,  which  is  suitable  for  use  in  charging  storage 
batteries  from  a  single-phase  circuit,  is  shown  with  its  con- 
nections in  Fig.  219.  It  consists  of  a  very  highly  exhausted 
glass  bulb  equipped  with  four  electrodes,  of  which  two 
are  positive,  one  negative,  and  the  other  an  auxiliary 
which  is  used  only  in  starting.  The  two  latter  electrodes 


CONVERTERS. 


297 


are  of  mercury.  The  two  external  terminals  of  an  auto- 
transformer  are  connected  with  the  two  positive  electrodes, 
while  the  internal  terminals  are  connected  to  the  single- 
phase  supply  circuit.  The  operation  of  this  converter  is 
based  upon  the  facts  that 
(a)  to  start  a  current  be- 
tween two  electrodes  in 
a  vacuum  bulb  of  this 
character  there  must  be 
impressed  upon  these 
electrodes  a  very  high 
voltage  (2  5, ooo  volts), most 
of  which  is  consumed  in 
overcoming  a  transition 
resistance  at  the  negative 
electrode,  and  (b)  once 
started  this  cathode  tran- 
sition voltage  drops  to  a 
very  small  value  (4  volts). 
In  operation,  and  after 
starting,  therefore,  current 
flows  during  one-half  of  a 
cycle  from  the  left-hand 
terminal  of  the  trans- 
former to  the  left-hand 
positive  electrode  through 
the  vapor  to  the  main 
negative  electrode  and  thence  through  the  battery  to  the 
center  of  the  transformer  coil,  and  during  the  following 
half  cycle  flows  from  the  right-hand  terminal  to  the  right- 
hand  positive  electrode  through  the  tube  and  battery  as 
before.  The  positive  electrodes  permit  current  to  flow 


Fig. 219. 


298  ALTERNATING-CURRENT    MACHINES. 

from  them  into  the  tube  but  never  in  the  reverse  direction. 
They  are  therefore  each  idle  during  alternate  half  cycles. 
The  transition  resistance  of  the  negative  main  electrode 
when  once  broken  down  remains  so  as  long  as  current 
enters  it  from  the  vapor. 

To  start  the  converter  the  bulb  is  tilted  until  there  is 
a  mercury  connection  between  the  main  negative  and 
auxiliary  electrodes.  This  permits  a  current  to  flow  from 
the  storage  battery  through  the  mercury  into  the  auxiliary 
electrode.  If  now  the  mercury  bridge  be  broken,  by 
restoring  the  bulb  to  its  original  position,  vapor  conduction 
will  be  established  between  the  main  negative  and  aux- 
iliary electrodes.  The  transition  resistance  of  the  latter 
is  thus  broken  down  and  the  converter  begins  to  operate, 
current  flowing  alternately  from  the  two  positive  electrodes 
to  the  auxiliary  electrode.  If  now  the  converter  be  again 
tilted  and  restored  to  its  normal  position  the  point  of 
entrance  of  the  vapor  current  into  the  mercury  can  be 
transferred  to  the  main  negative  electrode.  A  second 
tilting  is  seldom  necessary,  the  mercury  generally  making 
several  makes  and  breaks  of  the  circuit  during  the  first 
tilt  as  a  result  of  its  fluidity.  If  for  an  instant  (one  mil- 
lionth of  a  second)  the  current  ceases  to  enter  the  mercury, 
the  cathode  transition  resistance  will  reestablish  itself. 
An  inductance  inserted  in  the  battery  circuit  causes  a 
sufficient  lag  of  current  behind  the  voltage  between  a 
positive  and  the  negative  electrode  to  enable  the  voltage 
due  to  the  other  positive  electrode  to  maintain  the  opera- 
tion. The  current  in  the  battery  circuit  is  unidirectional 
but  pulsating. 


PROBLEMS.  299 


PROBLEMS. 

1.  From  what  points  on  the  armature  winding  should  taps  be  taken 
for  connection  with  the  successive  rings  of  a  5-ring  6-pole  converter? 

2.  A  4-pole  converter  is  supplied  with  six  slip-rings  so  as  to  be 
adapted  for  use  on  single-,  two-,  or  three-phase  circuits.     The  rings 
used  on  single-phase  are  i  and  4;  on  two-phase  are  i  and  4,  and  2  and 
6;  on  three-phase  i,  3,  and  5.     Locate  the  points  of  attachment  of  taps 
from  each  ring  to  the  armature  winding. 

3.  A  i2-ring  converter  delivers  600  volts  to  a  direct-current  railway 
circuit.     What  is  the  voltage  between  successive  slip-rings  ? 

4.  A  2o-pole  6-ring  converter  delivers  1000  amperes  of  direct  current 
at  full  load.    Neglecting  armature  resistance  and  other  losses,  determine 
the  current  wave-shape  in  a  conductor  20  electrical  degrees  in  advance 
of  tap  to  a  slip-ring. 

5.  During  ^vhat  portion  of  a  revolution  is  the  current  in  the  con- 
ductor mentioned  in  problem  4  so  directed  as  to  exert  a  motor  effort  ? 

6.  Compare  the  heating  effect  of  full-load  current  in  the  conductor 
of  problem  4  with  that  in  a  conductor  midway  between  taps. 


300  ALTERNATING-CURRENT   MACHINES. 


CHAPTER    IX. 

POWER   TRANSMISSION. 

114.  Superiority  of  Alternating  Currents.  —  In  trans- 
mitting power  electrically  over  long  distances,  it  is  neces- 
sary to  employ  high  voltages,  so  that,  with  a  reasonable 
line  loss,  the  cost  of  the  conductors  will  not  be  excessive. 
In  the  United  States,  power  transmission  at  high  voltages 
has  been  accomplished  by  means  of  alternating  current 
only.  In  Europe,  however,  considerable  attention  has 
been  given  to  the  development  of  the  Thury  system  of 
direct-current  transmission.  There  are  a  number  of 
plants  successfully  employing  this  system  at  the  present 
time,  but  the  highest  voltages  used  are  in  the  neighbor- 
hood of  20,000,  and  the  amounts  of  power  transmitted  are 
comparatively  small.  If  the  line  alone  be  considered, 
direct  current  is  far  superior  to  alternating  current.  The 
former  has  unity  power  factor,  is  free  from  inductive  dis- 
turbances, such  as  surges,  and  it  has  no  wattless  charging 
current  to  reduce  the  effective  output  of  the  machines. 
As  will  be  shown  later,  the  amount  of  conductor  material 
required  in  a  direct-current  line  is  less  than  that  required 
in  an  alternating-current  line  with  the  same  maximum 
voltage  in  the  two  cases. 

A  comparison  of  the  station  apparatus  of  both  systems 
of  power  transmission  shows  that  the  direct-current  system 
is  at  a  great  disadvantage.  Three  thousand  volts  is  the 


POWER  TRANSMISSION.  3OI 

maximum  that  can  be  successfully  handled  on  a  com- 
mutator, even  with  the  special  design  of  machine  such 
as  Thury  has  developed.  Consequently,  to  obtain  the 
required  high  line  voltage,  a  number  of  generators  must  be 
connected  in  series,  series-wound  machines  being  used. 
When  a  machine  is  generating  3000  volts,  the  maximum 
current  that  can  be  commutated  is  about  100  amperes,  so 
that  the  individual  machines  have  small  output.  Each 
generator  must  be  insulated  from  ground,  and,  as  several 
machines  are  connected  to  the  same  prime  mover,  they 
must  be  insulated  therefrom  and  from  each  other.  The 
system  is  grounded  at  the  middle  point,  so  as  to  limit  the 
amount  of  insulation  required;  that  is,  the  insulation 
under  each  machine  must  be  capable  of  withstanding  the 
maximum  difference  of  potential  between  its  terminals 
and  ground.  The  line  current  is  maintained  constant  by 
several  complicated  auxiliary  devices.  These  automatic- 
ally regulate  the  speed  of  all  the  prime  movers,  so  as  to 
keep  the  line  voltage  proportional  to  the  load;  cut  in  or 
out  of  circuit  one  or  more  machines  if  there  be  large 
changes  in  load,  and  short-circuit  any  disabled  machine. 

In  the  substations  a  number  of  series-wound  motors 
are  connected  in  series  across  the  line,  the  motors  being 
arranged  in  groups,  each  group  driving  a  generator.  The 
generators,  which  may  deliver  either  direct  or  alternating 
current,  are  connected  in  multiple  for  distribution.  The 
motors  and  generators  in  the  substation  must  be  insulated 
from  each  other  and  from  ground,  just  as  are  the  machines 
in  the  generating  station.  The  current  taken  by  the 
motors  is  kept  constant  by  an  automatic  shifting  of  the 
brushes.  The  Thury  system  is  adapted  only  to  under- 
takings where  the  power  is  to  be  transmitted  over  a  long 


302  ALTERNATING-CURRENT    MACHINES 

distance  and  the  load  is  to  be  concentrated  at  few  points 
since  at  every  tap  a  complete  substation  must  be  provided 
containing  motors  having  an  aggregate  voltage  equal  to 
the  line  voltage.  On  the  other  hand,  in  an  alternating- 
current  system,  a  static  transformer  can  be  installed  any- 
where along  the  line  and  it  will  operate  satisfactorily  with 
practically  no  attention. 

Considering  the  line,  alone,  the  employment  of  direct 
current  is  better  and  more  economical  than  that  of  alter- 
nating current.  But  when  the  whole  plant,  including 
generating  station,  line,  and  substations,  is  considered, 
the  employment  of  the  alternating-current  system  is  held 
by  many  engineers  to  be  the  most  advantageous.  The 
alternating-current  system  is  more  reliable,  more  flexible, 
and,  with  the  exception  of  special  cases,  is  probably 
cheaper  than  the  direct-current  system,  in  spite  of  the 
greater  cost  of  the  line  conductors. 

115.  Frequency.  —  According  to  the  Standardization 
Rules  of  the  A.  I.  E.  E.,  there  are  two  standard  fre- 
quencies, namely,  60  cycles  and  25  cycles.  In  early 
transmission  plants  the  frequency  employed  was  60  cycles 
or  higher.  All  recent  transmissions,  however,  are  at  25 
cycles,  and  there  is  a  strong  tendency  to  lower  this 
frequency  to  15  or  even  to  12?  for  certain  classes  of 
work.  Sixty-cycle  generators  and  transformers  are  smaller 
and  cheaper  than  are  those  of  lower  frequency.  It  was 
formerly  thought  that  for  lighting,  a  frequency  higher  than 
25  cycles  was  necessary  in  order  to  prevent  flickering  of 
the  lamps.  But  the  success  of  25-cycle  lighting  in  Buffalo 
from  the  circuits  of  the  Niagara  Falls  Power  Company 
has  proved  that,  if  the  form  factor  of  the  voltage  wave  is 


POWER  .TRANSMISSION.  303 

not  greater  than  that  for  a  sine  wave,  the  higher  frequency 
is  unnecessary.  The  Niagara  generators  give  a  wave 
slightly  flatter  than  a  sine  wave;  and  all  modern  genera- 
tors of  large  output  can  be  depended  upon  to  give  good 
wave  forms. 

The  advantages  of  low  frequency  for  transmission  lines 
are  as  follows:  (a)  The  inductive  drop,  2  xfLI,  is  less,  and 
consequently  the  regulation  is  better  than  for  high  fre- 
quencies. 

(b)  The   capacity   current,    2  xfEC,  also   increases  with 
the  frequency.     Its  effect  is  to  reduce  the  energy  output  of 
the  generators  and  transformers. 

(c)  The  lower  the  frequency,  the  less  difficult  becomes 
the  problem  of  operating  generators  and  other  synchronous 
apparatus  in   parallel.     This  is   because  the   unavoidable 
variations  in  speed  are  smaller  in  proportion  to  the  angular 
velocity,  the  lower  the  frequency. 

(d)  The  power  factor  of  an  induction  motor  decreases 
as  the  frequency  is  raised.     This  is  an  extremely  important 
reason  for  using  a  low  frequency,  since  the  power  load 
generally  constitutes  a  large  part  of  the  total  load  of  a 
transmission  system. 

(e)  A  low  frequency  is  also  less  liable  to  set  up  elec- 
trical  oscillations   as   a   result   of  the   coincidence   of  the 
natural   frequency  of  the   line  with  that  of  an  odd  har- 
monic of  the  impressed  E.M.F.     If  the  distributed  induc- 
tance and  capacity  of  the  line  be  L  henrys  and  C  farads 
respectively,    then    its    natural    frequency,    as    shown    by 
Steinmetz,  is  to  be  expressed  as 


/=          

~  4  VLC 


304  ALTERNATING-CURRENT   MACHINES. 

If  the  resistance  be  sufficiently  low,  as  is  often  the  case, 
oscillations  at  this  frequency  are  liable  to  occur.  A  triple 
harmonic  of  some  magnitude  usually  exists  in  the  E.M.F. 
wave  of  each  phase  winding  of  an  alternator.  This  does 
not  appear  at  the  terminals  of  a  three-phase  machine 
whether  Y-  or  A-connected.  It  does  appear,  however, 
between  the  terminals  and  a  grounded  neutral.  In  the 
armature  windings  there  is  usually  a  triple  harmonic  com- 
ponent of  current  which  sets  up  an  armature  reaction 
causing  magnetic  field  distortion  that  results  in  fifth  and 
seventh  harmonic  E.M.F's.  Triple  harmonics  of  E.M.F. 
or  of  current  also  result  from  the  use  of  transformers.  In 
three-phase  work  their  influence  upon  the  line  may  be 
overcome  by  the  use  of  A  connections. 

With  lines  constructed  in  accordance  with  present 
practice,  the  natural  frequency  for  a  length  of  150  miles  is 
about  three  hundred.  This  is  the  same  as  that  of  the 
fifth  harmonic  on  a  6o-cycle  system,  whereas  for  a  fre- 
quency of  25  the  fifth  harmonic  frequency  would  be  but 
125.  It  would  therefore  be  unwise  to  select  a  frequency  of 
60  cycles  for  such  a  line. 

116.  Number  of  Phases.  —  A  comparison  of  the  weights 
of  line  wire  of  a  given  material,  necessary  to  be  used  in 
transmitting  a  given  power,  at  a  given  loss,  over  the  same 
distance,  must  be  based  upon  equal  maximum  voltages 
between  the  wires.  For  the  losses  by  leakage,  the  thick- 
ness and  cost  of  insulation,  and  perhaps  the  risk  of  danger 
to  life,  are  dependent  upon  the  maximum  value.  A  com- 
parison upon  this  basis  gives,  according  to  Steinmetz,  the 
following  results:  — 


POWER   TRANSMISSION.  305 

Relative  weights  of  line  wire  to  transmit  equal  power  over 
the  same  distance  at  the  same  loss,  with  unit  power  factor. 

2  Wires.     Single-phase 100.0 

Continuous  current 50.0 

3  Wires.     Three-phase 75  .o 

Quarter-phase 145 . 7 

4  Wires.     Quarter-phase 100.0 

The  continuous  current  does  not  receive  the  approval 
of  American  engineers,  as  previously  stated.  The  single- 
phase  and  four-wire  quarter-phase  system  each  requires 
one-third  more  wire  than  the  three-phase  system. 

By  use  of  the  Scott  three-phase  quarter-phase  trans- 
former, the  transmission  system  may  be  three-phase, 
while  the  distribution  and  utilization  system  may  be 
quarter-phase. 

Each  conductor  of  a  three-phase  line  must  be  of  the  size 
required  in  a  single-phase  line  transmitting  half  as  much 
power,  with  the  same  percentage  of  loss,  at  the  same  volt- 
age and  distance  between  conductors. 

117.  Voltage.  —  If  the  frequency,  the  amount  of  trans- 
mitted power,  and  the  percentage  of  power  lost  in  the  line, 
remain  constant,  the  weight  of  line  wire  will  vary  inversely 
as  the  square  of  the  voltage  impressed  upon  the  line. 
This  depends  upon  the  fact  that  the  cross-section  of  the 
wire  is  not  determined  by  the  current  density  and  the  limit 
of  temperature  elevation,  but  by  the  permissible  voltage 
drop.  If  the  impressed  voltage  on  a  line  be  multiplied  by 
n,  the  drop  in  the  line  may  be  increased  n  times  without 
altering  the  line  loss.  For  the  line  loss  is  to  the  total 
power  given  to  the  line  as  the  drop  in  volts  is  to  the 


306  ALTERNATING-CURRENT    MACHINES. 

impressed  voltage.     To  transmit  the  same  power,  but  -  th 

n 

the  previous  current  is  necessary;  and  this  current,  to  pro- 
duce n  times  the  drop,  must,  therefore,  traverse  a  resistance 
n2  times  as  great  as  previously. 

In  a  long  transmission  line  the  conductors  constitute 
one  of  the  largest  items,  if  not  the  largest  item,  of  invest- 
ment of  the  entire  plant.  Consequently  it  is  desirable  to 
have  the  voltage  as  high  as  possible.  But  raising  the 
voltage  increases  the  investment  for  transformers,  switch- 
ing apparatus,  lightning  protection,  and  insulators;  and  the 
depreciation  and  repair  charges  on  these  items  are  much 
greater  than  the  corresponding  charges  on  the  conductors. 
The  economic  voltage  to  be  employed  for  transmitting  a 
given  amount  of  power  over  a  certain  distance  is  that 
voltage  which  will  lead  to  the  minimum  annual  cost  for 
the  entire  plant.  Theoretically,  this  economic  voltage  can 
be  determined  by  expressing  the  several  elements  of  cost 
as  functions  of  the  voltage  and  equating  the  differential 
of  this  expression  to  zero.  This  method  is  complicated, 
and  it  requires  so  many  assumptions  as  to  render  it  of 
little  use. 

118.  Economic  Drop.  —  A  more  practical  way  of  deter- 
mining the  voltage  is  based  upon  the  fact  that  there  are 
certain  standard  voltages  for  high-tension  transformers. 
Except  for  special  cases,  a  standard  voltage  should  be 
used.  For  a  given  voltage,  the  amount  of  conductor 
material  varies  inversely  as  the  drop,  whereas  the  line  loss 
varies  directly  with  the  drop.  If  the  economic  drop, 
which  fixes  the  cross-section  of  the  conductor,  be  calcu- 
lated for  the  several  standard  voltages,  the  best  voltage  to 


POWER  TRANSMISSION.  307 

employ  can  readily  be  determined.  For  a  given  voltage 
at  the  generating  station,  the  economic  drop  and  cross- 
section  of  conductor  for  a  single-phase  circuit  may  be 
found  as  follows: 

Let     E    =  voltage  at  generating  station, 

P    =  power  in  kilowatts  at  generating  station, 

Zt  =  length  of  line  in  miles,  i.e.,  length  of  a  single 

conductor, 

R    =  total  resistance  of  line  in  ohms, 
x    =  loss  in  terms  of  impressed  quantities, 
5    =  section  of  conductor  in  circular  mils, 
Cl  =  cost  of  energy  in  dollars   per  kilowatt-year  at 

generating  station, 

c2  =  cost  of  conductor  in  dollars  per  pound, 
p2  =  interest  rate  on  cost  of  line  conductors, 
K^  =  resistance  in  ohms  per  mile  of  conductor  hav- 
ing one  circular  mil  cross-section,  and 
K2  =  weight  in  pounds  per  mile  of  conductor  having 
one  circular  mil  cross-section. 

Then,  line  loss  =  Px. 

Annual  cost  of  line  loss  =  cfx. 
Weight  of  line  conductors  =  2  K2L^S. 
Cost  of  line  conductors  =  2  c2K2L±S. 
Annual  cost  of  line  conductors  =  2  p^c2K2LvS. 

2  L 
Line  resistance  =  R  =  K^  — — -  • 

o 

T .       ,              ,.,         1000  P  D        2000  PKJL.^ 
Line  drop  =  Ex  =  — —  R  = ^S~^ 

Section  of  conductor  =  S  =  — ^—  K^ — L« 


308  ALTERNATING-CURRENT  MACHINES. 

The  total  annual  charge  due  to  line  loss  plus  interest  on 
conductors  is 

cfx  +  2  p&KzLiS, 
and  per  delivered  kilowatt  is 

c^Px  +  2  p,c2K,L,S 
~ 


~  P  -Px 

Substituting  the  value  of  S,  this  becomes 


or  _gg_ftc.:.:.  4V  LOOP.  () 

9      i  -  x  E2x  (i  -  x) 

If  K  is  substituted  for  p^KtKt  4  i,2  1000,  then 

c,ar  K 

9  =  7^^  +  £>»  (i  -  *j  '  (4) 

To  find  the  minimum  value  of  q,  its  derivative  is  placed 
equal  to  zero,  and  there  obtains 

dq  2  K  K 

=  V2+r*-2=o;  (5) 


whence         *  =  -  --  ±          -    +       -     . 
£2ti  *  ^ 

But  as  jc  is  positive 


If  the  preceding  were  worked  out  for  a  constant  or  fixed 
delivered  E.M.F.,  instead  of  a  fixed  impressed  E.M.F., 
allowing  the  latter  to  become  what  it  might,  the  expression 


POWER  TRANSMISSION.  309 

for  q  would  be  the  same,  except  that  the  denominator 
would  be  P  instead  of  P  (i  —  x),  the  quantities  E,  x,  P, 
etc.,  being  then  delivered  quantities  instead  of  impressed 
quantities.  If  this  be  done,  and  the  value  of  q  be  differ- 
entiated, there  obtains 

dq__  K 

dx  ~  °l      EV 
Hence 

TT 

—  —  =  c^x,  that  is,  the  well-known  relation 

Interest  =  Loss. 

A  three-phase  line  requires  three-quarters  as  much  con- 
ductor material  as  a  single-phase  line  transmitting  the 
same  amount  of  power  with  the  same  loss.  Each  con- 
ductor of  a  three-phase  transmission  line  has  one-half  the 
area  of  each  conductor  of  the  equivalent  single-phase  line. 
To  find  the  economic  drop  for  a  three-phase  line,  multiply 
2  pzCzKz^S  in  equation  (i)  by  f.  Then  f  K  will  appear 
in  equation  (4)  instead  of  K.  Solving  for  the  economic 
drop, 


The  area  of  each  conductor  is 

s.if^.ii).      „, 

1  19.  Line  Resistance.  —  The  resistance  of  anything  but 
very  large  lines  is  the  same  for  alternating  currents  as  for 
direct  currents.  In  the  larger  sizes,  however,  the  resist- 
ance is  greater  for  the  alternating  currents.  The  reason 
for  the  increase  is  the  fact  that  the  current  density  is  not 


310 


ALTERNATING-CURRENT   MACHINES. 


uniform  throughout  a  cross-section  of  the  conductor,  but 
is  greater  toward  its  outside.  The  lack  of  uniformity  of 
density  is  due  to  counter  electromotive  forces  set  up,  in 


/ 

INCREASE  OF  RESISTANCE 
PERCENT 

4»  «o  io  <*  C 

/ 

/ 

/ 

/ 

/ 

<s 

/ 

/s 

:  * 

^^ 

^ 

3    10     20    .30     49     50    60     70     80    90    10 

CIRCULAR  MILS   X  FREQUENCY 
Fig.  220. 

the  interior  of  the  wire,  by  the  varying  flux  around  the 
axis  of  the  wire  which  accompanies  the  alternations  of  the 
current.  This  phenomenon  is  termed  skin  effect,  §21.  Its 
magnitude  may  be  determined  from  the  curve,  Fig.  220. 

120.  Line  Inductance.  —  The  varying  flux  which  is  set 
up  between  the  two  line  wires  of  a  single-phase  trans- 
mission circuit  by  the  current  flowing  in  them  gives  rise 
to  a  self-induced  counter  E.M.F.  The  inductance  per 
unit  length  of  single  wire  is  numerically  equal  to  the  .flux 
per  unit  current,  which  links  a  unit  length  of  the  line. 
To  determine  this  value  consider  a  single-phase  line,  with 
wires  of  R  cms.  radius,  strung  with  d  cms.  between  their 
centers,  and  carrying  a  current  i.  Let  a  cross-section  of 


POWER  TRANSMISSION. 


311 


the  line  be  represented  in  Fig.  221.     The  flux  d^^  which 
passes  through  an  element  dr  wide  and  of  unit  length,  is 


Fig.  221. 

equal  to  the  magnetomotive  force  divided   by  the  reluc- 
tance, or 


2  nr 
dr 

Integrating  for  values  of  r  between  d  ~  R  and  R, 

~  R\ 


Fig.  222. 


and  practically  =  2  i  log  f  -  j  . 

There  is  some  flux  which  surrounds 
the  axis  of  the  right-hand  wire,  and 
which  lies  inside  the  metal.  This  is  of 
appreciable  magnitude  owing  to  the 
greater  flux  density  near  the  wire. 
Represent  the  wire  by  the  circle  in 
p^  222,  and  suppose  that  the  current 
is  uniformly  distributed  over  the  wire.  Then  the  current 


312  ALTERNATING-CURRENT  MACHINES. 

y? 
inside  the  circle  of  radius  x  is   —  i,  and  the   magneto- 

motive force  which  it  produces  is 


The   flux,  however,   which  it    produces  links  itself  with 

oc2 
but  -^ths  of  the  wire.     The  flux  through  the  element  dx, 

which  can  be  considered  as  linking  the  circuit,  is  therefore 

,,        2  x3i  dx 
d<&^  =  -        -  . 
I*R* 

Integrating  for  values  of  x  between  o  and  R, 


AM 

For  copper  or  aluminum  wires  /*  =  i.    Hence   the   total 
flux  linked  with  the  line  is 


and  the  inductance,  in  absolute  units,  being  the  flux  per 
unit  current,  is 


This  gives  by  reduction  the  inductance  in  henrys  per  wire 
per  mile  as 

L  =[80.5  +  740  log  (I)]  io- 6. 

The  drop  in  volts  due  to  the  inductance,  per  mile  of  line 
(two  conductors)  per  unit  current,  is  therefore 


POWER  TRANSMISSION. 


313 


EL  =  .00405/2.3  log  ^ 

where  d  and  R  must  be  in  terms  of  the  same  unit. 

It  will  be  noted  that  the  inductance  depends  upon 
the  distance  between  conductors.  This  distance  should 
increase  with  the  voltage,  but  there  is  no  definite  relation 
between  them.  The  following  values  represent  average 
practice  for  bare  overhead  conductors: 


Kilovolts. 

Distance  between  Con- 
ductors in  Inches. 

2.3  to  6.6 

28 

10  to  20 

40 

20  to  30 

3°  to  5° 
50  to  60 

48 
60 

72 

121.  Line  Capacity.  —  The  two  conductors  of  a  single- 
phase  transmission  line,  together  with  the  air  between 
them,  act  as  a  condenser.  The  conductors  correspond  to 
the  condenser  plates,  and  the  air  corresponds  to  the  dielec- 
tric. When  the  lines  are  long,  or  when  the  conduc|ors  are 
close  together,  the  capacity  is  quite  appreciable. 

The  capacity  between  two  parallel  cylindrical  con- 
ductors may  be  determined  as  follows:  Let  A  and  B 
(Fig.  223)  represent  the  two  conductors  of  R  cms.  radius 
and  d  cms.  apart  between  centers.  Let  A  be  charged  with 
+  Q  electrostatic  units  of  electricity  per  centimeter  of  length, 
and  B  with  —  Q  units.  If  the  charge  on  A  be  alone  con- 
sidered, there  emanates  from  each  unit  length  an  electro- 
static flux  of  4  7r<2  lines  directed  radially  away  from  the  axis 
of  A.  Similarly,  a  flux  of  —  4  nQ  emanates  radially  from 
each  unit  length  of  B,  due  to  its  charge.  The  negative  sign 


314 


ALTERNATING-CURRENT    MACHINES. 


indicates  that  the  flux  is  directed  towards  the  axis  of  B. 
The  superposition  of  these  two  fluxes  results  in  an  electro- 
static field  such  as  would  exist  if  a  neutral  conducting 


NEUTRAL 
PLANE 


Fig.  223. 

plane  were  introduced  halfway  between  A  and  B  and 
perpendicular  to  the  plane  of  their  axes,  and  the  potential 
of  the  neutral  conducting  plane  were  maintained  as  much 
below  that  of  A  as  it  is  above  that  of  B.  To  determine 
the  potential  differences,  consider  that  the  difference  of 
potential  between  two  points  is  equal  to  the  work  that 
must  be  performed  on  a  unit  positive  charge  to  move  it 
from  one  point  to  the  other.  The  intensity  of  the  field  at  a 
point  C,  at  a  distance  x  from  the  axis  of  A,  due  to  the 
charge  on  A,  this  intensity  being  the  flux-density  or  force 
which  would  be  exerted  upon  a  unit  positive  charge,  is 


2  TtX 


and  that  due  to  the  charge  on  5,  noting  that  it  is  in  the 
same  direction  as  that  due  to  A,  is 


2  n  (d  —  x)       d  —  x 


POWER  TRANSMISSION.  31$ 

Therefore  the  total  intensity  or  force  is 


Hence  the  difference  of  potential  between  A  and  the 
neutral  plane,  which  is  the  same  as  that  between  the 
neutral  plane  and  B,  is 


The  capacity  between  either  conductor  and  the  neutral 
plane  is  therefore 

C  =      ==~  9 


in  electrostatic  units  per  centimeter  length  of  conductor. 
In  transmission  lines,  R  is  usually  so  small  compared  with 
d  as  to  be  neglected.  Reducing  to  miles,  microfarads,  and 
common  logarithms,  the  capacity  between  neutral  plane 
and  either  conductor  is 

„       0.0388  £       ,  ., 

C  =  -  -  —  -  microfarads  per  mile, 


and  between  conductors  it  is  half  as  much. 

Because  of  its  capacity,  a  line  takes  a  charging  current 
when  an  alternating  E.M.F.  is  impressed  upon  it,  even 
though  it  be  not  connected  to  a  load.  The  value  of  this 
current  is 

1  =  2  xfEC  io~  6  amperes, 

where  E  and  C  are  the  voltage  and  the  capacity  in  micro- 


3l6  ALTERNATING-CURRENT   MACHINES. 

farads   respectively   between   conductors   or  between    one 
conductor  and  the  neutral  plane. 

To  determine  the  charging  current  per  conductor  of  a 
three-phase  line,  the  above  formula  is  used,  C  and  E  being 
the  capacity  and  voltage  respectively  between  conductor 

and  neutral  plane.     The  voltage  is   — - =  of  that  between 

^3 
conductors. 

The  capacity  between  a  conductor  and  ground  may  be 
derived  by  considering  the  ground  as  the  neutral  plane. 
In  a  circuit  not  employing  a  ground  return,  there  is  no 
charging  current  due  to  the  capacity  between  the  con- 
ductors and  ground,  but  in  the  case  of  a  single  conductor 
with  ground  return  there  is  a  charging  current.  The 
value  of  this  current  is  given  by  the  preceding  formula,  in 
which  case  C  is  the  capacity  to  ground  and  E  is  the  voltage 
to  ground. 

122.  Regulation.  —  The  regulation  of  a  transmission 
line  is  the  ratio  of  the  maximum  voltage  difference  at  the 
receiving  end,  between  rated  non-inductive  load  and  no- 
load,  to  the  rated-load  voltage  at  the  receiving  end,  con- 
stant voltage  being  impressed  upon  the  sending  end. 

In  short  aerial  transmission  lines,  the  capacity  and 
charging  current  may  be  considered  as  negligible.  The 
voltage  at  the  receiving  end  will  then  be  the  same  as  that 
at  the  sending  end  on  no-load.  On  full-load  the  sending 
voltage  is  equal  to  the  vectorial  sum  of  the  delivered 
voltage,  that  necessary  to  overcome  the  resistance  drop, 
and  that  necessary  to  overcome  the  inductive  drop  at  90° 
ahead  of  the  delivered  voltage. 

Ill    long    transmission    lines    the    capacity    cannot  be 


POWER   TRANSMISSION,  317 

neglected,  and  the  *  folio  wing  method  due  to  Steinmetz 
may  be  employed:  Consider  the  line  to  be  made  up  of  a 
number  of  sections,  say  ten,  to  each  of  which  is  appor- 
tioned one-tenth  the  total  capacity  and  inductance  of  the 
line.  The  capacities  may  be  considered  as  localized 
condensers  at  the  sending  end  of  the  section  and 
connected  across  the  lines.  The  inductances  may  be 
considered  as  connected  in  series  with  the  line.  The  con- 
nections of  a  few  sections  are  shown  in  Fig.  224.  The 
inductive  and  resistance  drop  of  voltage  in  each  section  is 


A        A»GVClVlil£ 

•J— (Tjwffx-l—afi5fflr> 


Fig.  224. 


greater  than  in  any  other  section  more  remote  from  the 
sending  end,  because,  although  the  inductances  and 
resistances  are  the  same  in  all  sections,  the  current  is 
greater  as  a  result  of  the  extra  charging  current  due  to 
the  capacity  of  intervening  sections.  The  charging  cur- 
rent is  also  different  for  each  section  because  the  voltage 
which  occasions  it  increases  as  the  sending  end  is 
approached. 

The  voltages  and  currents  in  each  section  can  be 
determined  with  sufficient  accuracy  by  making  use  of  a 
large  vector  diagram,  such  as  indicated  in  Fig.  225,  where 
OE  and  OI  represent  the  delivered  voltage  and  current 
respectively.  The  power  factor  of  the  load  being  unity,  the 
latter  are  in  phase  with  each  other.  Let  E,  El}  E2,  ...  be 
the  voltages,  and  /,  Ily  72,  .  .  .  be  the  currents  delivered 
to  the  load  and  the  successive  sections  respectively.  Then, 


ALTERNATING-CURRENT   MACHINES. 


if  R  and  X  be  the  resistance  and  inductive  reactance 
in  ohms,  and  C  be  the  capacity  in  farads,  of  each  and 
every  section, 

El=  (E  +  RI)  0  XI  at  90°  lead, 
E2  =  (Ei  +  RIJ  0  XIi  at  90°  lead,  etc., 
and  /!  =  /  0  ajE^C  at  90°  lead, 

1 2  =  A  0  wE2C  at  9°°  lead,  etc. 

The  various  phase  relations  and  magnitudes  are  seen  in 
the  figure,  wrhere  the  E's  and  JPs  with  various  subscripts 

0  .IR . 

rr^: — • — il 

w  EtC 

L  E2C 


E' 


Fig.  225. 


mark  the  terminals  of  the  vectors  from  the  origin,  which 
are  not  drawn  for  the  sake  of  clearness. 

The  cosine  of  <£/,,  the  angle  between  the  current  /'  and 
the  voltage  Ef  at  the  sending  end  of  the  line,  is  the  power 
factor  of  the  line. 

This  method  is  strictly  accurate  only  when  the  number 
of  sections  is  infinite.  Ten  sections  give  sufficient  accu- 
racy for  practical  work. 

123.  Conductor  Material.  —  The  high  permeability  of 
iron  prohibits  its  use  as  a  conductor  for  transmission 


POWER  TRANSMISSION. 


319 


lines.     There  are  but  two  other  materials  available,  copper 
and  aluminum.     The  physical  constants  of  these  metals 


-•'•*' 

Copper. 

Aluminum. 

Specific  gravity 

8,07 

2   68 

Conductivity,  in  terms  of  Matthiessen's  Standard  .... 
Tensile  strength   pounds  per  square  inch  ...   ... 

.98 

60,000 

.61 

26  ooo 

Elastic  limit    pounds  per  square  inch 

40  ooo 

14  ooo 

Stretch  modulus  of  elasticity,  pounds  per  square  inch 
Coefficient  of  expansion  per  degree  Fahrenheit.  .  .  .  . 

16,000,000 
0.0000096 

9,000,000 
0.0000128 

For    the    same    conductivity,    an    aluminum    conductor 

must  have  -f-  or  1.6  times  the  area  of  a  copper  conductor. 
.61 

But  as  aluminum  is  three-tenths  as  heavy  as  copper,  the 
former  weighs  but  0.48  as  much  as  the  latter  for  equal 
conductivities.  Therefore,  if  aluminum  costs  less  than 
2.08  times  as  much  as  copper  per  unit  of  weight,  it  is 
cheaper  than  the  latter.  The  prices  of  both  metals  vary, 
but  that  of  aluminum  is  usually  much  less  than  2.08  times 
that  of  copper.  If  its  lower  tensile  strength  and  greater 
coefficient  of  expansion  are  properly  allowed  for  while  the 
line  is  being  strung,  the  use  of  aluminum  for  transmission 
line  conductors  is  just  about  as  satisfactory  as  that  of 
copper.  Because  of  the  resultant  saving,  aluminum  is 
being  used  very  extensively. 

124.  Insulators.  —  There  is  no  material  from  which 
insulators  can  be  made  that  possesses  all  the  mechanical 
and  electrical  qualities  to  be  desired.  Glass  has  been  used 
to  some  extent,  but  it  is  easily  broken,  either  in  transit,  or 


320  ALTERNATING-CURRENT   MACHINES. 

by  stones  and  bullets  after  the  insulators  are  installed. 
The  two  materials  now  used  for  insulators  on  high-tension 
transmission  lines  are  porcelain  and  a  substance  known  as 
electrose. 

Porcelain  is  much  tougher  than  glass  and  is  therefore 
not  so  liable  to  be  broken.  Porcelain  insulators  are 
heavily  glazed  to  prevent  absorption  of  moisture,  since 


Fig.  226. 

even  the  best  porcelain  is  somewhat  porous.  A  brown  or 
gray  tint  is  usually  introduced  into  the  glaze  so  that  the 
insulators  will  not  attract  the  attention  of  marksmen.  A 
Thomas  33,ooo-volt  porcelain  pin-type  insulator  is  shown 
in  Fig.  226.  It  is  y|  inches  high  and  8J  inches  in  diameter. 
Electrose  possesses  good  insulating  qualities,  is  very 
strong  mechanically,  and  is  free  from  cracks.  It  has  a 
brown,  smooth,  polished  surface  and  does  not  absorb 


POWER  TRANSMISSION. 


321 


moisture.  Metal  parts  may  be  molded  into  it  readily  if 
so  required.  A  24,ooo-volt  electrose  pin-type  insulator  is 
shown* in  section  in  Fig.  227.  It  is  7  inches  high  and  12 
inches  in  diameter. 

In  designing  an  insulator,  the  distance  along  the  surface 


from  the  conductor  to  the  point  of  support  should  be  as 
long  as  possible  in  order  to  decrease  the  leakage  current. 
The  shortest  distance  through  the  air  between  these  two 
points  should  be  great  enough  to  prevent  the  line  voltage 
flashing  over,  even  when  the  insulator  is  wet.  Further- 
more the  distance  through  the  dielectric  should  be  great 


322  ALTERNATING-CURRENT   MACHINES. 

enough  to  prevent  puncture.  At  the  same  time  the  size  of 
the  insulator  and  the  quantity  of  material  used  should  be 
kept  as  small  .as  possible.  In  view  of  these  requirements 
high-voltage  insulators  have  taken  the  form  of  a  series 
of  umbrella-shaped  petticoats.  t  Manufacturing  difficulties 
prevent  the  construction  of  a  satisfactory  large  porcelain 
insulator  of  the  usual  type  in  a  single  piece.  The  petti- 
coats therefore  are  made  separately  and  are  glazed  or 
cemented  together,  this  being  usually  done  where  the  line 
is  being  erected.  Electrose  insulators  may  be  molded  in 
one  piece,  regardless  of  shape. 

Metal  pins  are  generally  used  for  pin-type  insulators, 
even  on  wooden  poles,  because  of  their  strength  and  the 
fact  that  they  are  not  burned  by  the  leakage  current.  It 
was  formerly  thought  that  the  additional  insulation  of  a 
wooden  pin  was  desirable,  but  it  has  been  found  that  even 
the  best  treated  pins  will  absorb  moisture  in  time,  especially 
in  salt  atmospheres.  With  steel  towers  and  long  spans, 
the  large  strains  on  the  insulators  necessitate  the  use  of 
heavy  iron  or  steel  pins. 

The  tendency  toward  higher  voltages  for  power  trans- 
mission has  led  to  the  application  of  flexible  suspension- 
type  insulators.  These  consist  of  individual  units  securely 
coupled  together,  the  number  of  units  to  be  employed 
depending  upon  the  line  voltage.  Some  no,ooo-volt 
insulators  of  this  type  are  shown  attached  to  a  trans- 
mission tower  in  Fig.  230. 

125.  Sag  of  Conductors.  —  In  stringing  the  conductors 
there  are  two  things  to  be  taken  into  consideration. 
First,  the  greatest  possible  tension  in  the  conductor,  which 
will  occur  at  minimum  temperature,  must  be  less  than  the 


POWER  TRANSMISSION.  323 

elastic  limit  of  the  conductor.  At  the  minimum  tempera- 
ture there  may  be  a  coating  of  ice  on  the  conductor.  This 
not  only  results  in  increased  weight,  but  also  presents  a 
greater  area  to  the  wind.  Second,  the  clearance  between 
conductors  and  ground  at  the  maximum  temperature 
must  be  great  enough  to  prevent  accidental  contact  or 
malicious  interference.  These  questions  are  of  greater 
importance  when  using  aluminum  than  when  using  copper, 
because  the  tensile  strength  of  aluminum  is  less  than  that 
of  copper,  its  coefficient  of  expansion  is  greater,  and,  for  a 
given  conductivity,  it  presents  a  larger  surface  to  the  wind. 
The  conductors  are  of  necessity  strung  under  varying 
conditions  as  to  temperature,  and  consequently  they 
should  be  given  an  appropriate  sag  so  that  at  the  extreme 
temperatures  the  required  conditions  will  be  fulfilled.  It 
is  common  to  make  use  of  curves,  one  for  each  span 
length,  whose  ordinates  represent  the  appropriate  sag  and 
whose  abscissae  represent  temperatures  between,  say, 
—  40°  and  no°F.  There  is  considerable  difference  of 
opinion  as  to  the  proper  assumptions  to  be  made  for  sleet 
and  wind  pressure.  In  northern  countries,  conductors 
will  frequently  be  covered  with  ice  from  one-half  to  one 
inch  thick  all  around,  and  hence  this  should  be  allowed 
for.  In  regard  to  wind  pressure,  it  should  be  noted  that 
the  wind  velocities  published  by  the  United  States  Weather 
Bureau  are  observed  and  not  actual  velocities.  An 
observed  velocity  of  100  miles  per  hour  corresponds  to  an 
actual  velocity  of  about  75  miles  per  hour.  The  wind 
pressure  in  pounds  per  square  foot  exerted  upon  a  plane 
surface  normal  to  the  direction  of  the  wind  may  be 
expressed  as  CV2,  where  V  is  the  actual  wind  velocity  in 
miles  per  hour  and  C  is  a  constant  whose  value  may  be 


324 


ALTERNATING-CURRENT  MACHINES. 


taken  as  .005.  Thus  the  wind  pressure  on  a  plane  normal 
surface  for  an  actual  velocity  of  75  miles  per  hour  is 
approximately  30  pounds  per  square  foot.  The  wind 
pressure  on  the  conductors  is  usually  taken  as  half  of  this 
value,  or  15  pounds  per  square  foot  of  projected  conductor 
area.  Except  in  the  case  of  tornadoes,  observed  wind 
velocities  in  excess  of  100  miles  per  hour  are  practically 
unknown.  The  assumption  of  a  wind  pressure  of  15 
pounds  per  square  foot  of  projected  area  when  the  con- 
ductor is  covered  with  one-half  inch  of  ice  all  around  is 
conservative. 

The  weight  of  the  conductor  and  ice  acts  vertically 
downward,  while  the  wind  pressure  at  worst  acts  trans- 
versely to  the  direction  of  the  line.  The  sag  of  the  con- 
ductor is  therefore  in  the  direction  of  the  resultant  of  these 


two  forces,  as  shown  in  Fig.  228.     The  appropriate  sag  of 
a  conductor  at  any  given  temperature  such  that  the  elastic 


POWER   TRANSMISSION.  325 

limit  of  the  metal  shall  not  be  exceeded  at  the  minimum 
temperature  may  be  found  in  the  following  manner: 
Let      St  =  span  in  feet  (Fig.  229), 
D    =  sag  in  feet  (Fig.  228), 

W   =  weight  of  conductor  and  ice  in  pounds  per  foot, 
Wr  =  resultant  of  W  and  wind  pressure  in  pounds 

per  foot, 
T   =  maximum  allowable  tension  in  the  conductor, 

usually  taken  as  the  elastic  limit, 
t   =  temperature  in  degrees  Fahr.  above  the  mini- 
mum (  —  40°), 
k   =  temperature  coefficient  of  linear  expansion  per 

degree  F. 

A    =  cross-section  of  conductor  in  square  inches, 
E  ±=  stretch   modulus  of    elasticity   in    pound -inch 

units, 
Ls  =  length  of  single   span  of  strung  cable   at   the 

minimum  temperature  in  feet,  and 
Lu=  length    of  unstressed   single  span  of  cable  at 
minimum  temperature. 


Fig.  229. 

Then  the  following  relations  are  sufficiently  exact: 


326  ALTERNATING-CURRENT   MACHINES. 

L. 

Lu-  -j-, 


from  which  D,  the  sag  in  the  direction  indicated  in  Fig.  228. 

may  be  found.    The  vertical  sag,  D',  is  equal  to  -—  . 

Wr 

126.  Line  Structure.  —  There  are  two  types  of  line 
structure  for  transmission  lines  carrying  large  amounts  of 
power  at  high  voltages,  namely,  wooden  poles  and  steel 
towers.  If  wooden  poles  are  used  the  spans  must  be 
short,  in  order  that  the  poles  may  withstand  the  forces  to 
which  they  are  subjected.  With  short  spans,  the  expense 
for  insulators  will  be  large.  On  the  other  hand,  while  a 
single  steel  tower  costs  a  great  deal  more  than  a  wooden 
pole,  the  use  of  towers  permits  of  longer  spans  and 
thereby  reduces  the  number  of  structures  and  the  cost  of 
insulators.  The  depreciation  and  repair  charges  for  steel 
towers  are  very  small  compared  with  the  same  items  for 
wooden  poles,  especially  if  the  towers  be  galvanized. 
Taking  account  of  interest  on  the  investment  and  the 
depreciation  and  repair  charges  on  the  line  structures, 
including  insulators,  it  will  usually  be  found  that  towers 
are  cheaper  than  wooden  poles  for  lines  using  heavy  con- 
ductors. Even  if  the  cost  of  a  proposed  line  with  towers 
is  a  little  more  than  with  poles,  towers  would  nevertheless 
be  used  in  most  cases  because  of  their  greater  reliability. 
Towers  can  be  designed  to  withstand  the  maximum  forces 
which  will  be  exerted  upon  them,  allowing  any  desired 


POWER   TRANSMISSION.  327 


Fig.  230. 


328  ALTERNATING-CURRENT   MACHINES. 

factor  of  safety.  The  strength  of  a  tower  will  remain 
constant,  whereas  the  strength  of  poles  of  a  given  size 
and  kind  of  wood  will  vary,  and  the  original  strength  will 
gradually  diminish  until  the  poles  fail. 

Fig.  230  shows  the  transmission  line  tower  used  by  the 
Hydro-Electric  Power  Commission  of  Ontario,  Canada,  in 
transmitting  power  from  Niagara  to  various  points  in  that 
province  at  110,000  volts  and  25  cycles.  The  line  con- 
sists of  two  three-phase  circuits,  the  cables  of  each  circuit 
being  nine  feet  apart.  The  tower  is  60  feet  high,  with  a 
base  1 6  feet  square,  the  span  being  550  feet.  Lightning 
protection  is  secured  by  the  use  of  overhead  ground  wires 
fastened  to  the  tower,  as  shown. 

The  forces  acting  on  a  structure  in  a  straight  portion  of 
the  line,  where  the  spans  are  of  equal  length,  are:  (a)  the 
weight  of  the  conductors  and  ice;  (b)  the  wind  pressure 
on  the  conductors  when  covered  with  ice;  (c)  the  wind 
pressure  on  the  structure;  (d)  the  weight  of  the  structure. 

The  method  of  calculating  the  first  three  forces  has  been 
considered.  The  first  and  last  act  vertically  downward, 
while  the  others  are  considered  as  acting  transversely  to 
the  direction  of  the  line.  The  value  of  the  wind  pressure 
is  a  maximum  when  the  direction  of  the  wind  is  trans- 
verse to  that  of  the  line,  and  zero  when  the  wind  is  in  the 
same  direction  as  the  line.  Theoretically,  there  is  no 
force  on  the  structure  due  to  the  tension  in  the  conductors, 
since  the  tensions  in  two  adjacent  spans  counterbalance 
each  other.  In  practice,  however,  such  a  force  exists,  on 
account  of  differences  of  wind  pressure,  of  elevation  of  the 
towers,  and  of  length  of  spans. 

Should  one  or  more  of  the  conductors  break,  the  full 
tension  thereof  will  come  on  the  structures  on  each  side  of 


POWER   TRANSMISSION.  329 

the  break.  Such  an  accident  is  unusual,  and  the  addi- 
tional expense  of  building  each  structure  to  withstand  it 
would  not  be  justified.  Consequently  the  conductors  are 
secured  to  the  insulators  by  means  of  loose  ties.  In  the 
event  of  the  breaking  of  a  conductor  the  tension  will  then 
be  taken  up  by  several  structures.  At  intervals  of  a  few 
miles  there  are  guyed  towers  capable  of  taking  the  full 
tension,  and  the  conductors  are  firmly  clamped  to  the 
insulators  on  such  structures. 

Knowing  the  magnitude  of  the  transverse  forces,  the 
strength  of  the  pole  required  to  resist  them  can  readily  be 
calculated  by  means  of  the  well-known  formula  for  a  beam 
fixed  at  one  end.  The  resisting  moment 

SI 
M==-, 

where  S  is  the  stress  in  the  section,  7  is  the  moment  of 
inertia  of  the  section,  and  C  is  the  distance  from  center  to 
the  fiber  under  maximum  stress.  From  this  formula,  the 
most  economical  wooden  pole  is  one  whose  vertical  sec- 
tion is  a  parabola.  The  strength  of  the  pole  necessary 
to  resist  the  other  two  forces  can  be  determined  by  means 
of  a  formula  for  the  compressive  strength  of  a  long 
column.  However,  if  a  structure  is  strong  enough  to 
resist  the  transverse  forces,  it  will  undoubtedly  be  strong 
enough  to  resist  the  vertical  forces. 

127.  Spans  and  Layout.  —  For  a  given  sized  conductor, 
the  necessary  heights  of  the  line  structure,  and  the  loads 
which  the  latter  must  sustain,  increase  as  the  span  is 
lengthened.  Consequently  the  cost  of  a  single  structure 
increases  as  the  span  is  increased.  At  the  same  time, 
however,  the  number  of  structures  and  of  insulators 


330          ALTERNATING-CURRENT   MACHINES. 


required  is  diminished.  If  the  total  cost  of  the  structures 
complete  with  insulators  be  calculated  for  different  spans, 
the  most  economical  length  of  span  can  be  determined. 
The  foregoing  applies  to  either  wooden  poles  or  steel 
towers,  but  of  course  poles  can  be  used  only  for  com- 
paratively short  spans  because  of  strength  limitations. 

On  account  of  irregularities  in  the  ground  and  in  order 
to  clear  obstacles,  spans  longer  or  shorter  than  the  stand- 
ard are  frequently  necessary.  In  such  cases  the  forces 
acting  on  the  structures  are  other  than  normal.  These 
may  sometimes  be  taken  care  of  by  using  the  regular 
structures  with  guys,  but  otherwise  special  structures  are 
necessary. 

In  straight  portions  of  the  line  ordinarily  there  are  no 
forces  acting  on  the  structures  due  to  the  tension  in  the 

conductors,  since  the 
tension  on  one  side 
balances  that  on  the 
other.  But  when  the 
line  changes  in  direction 

this  is  not  so.  Figs.  231  and  232  show  two 
methods  of  making  a  change  in  direction. 
In  the  first,  the  conductors  are  dead-ended 
on  the  two  structures,  and  the  tension  is  taken 
up  by  guys.  In  Fig.  232,  for  simplicity, 
but  one  conductor  is  shown.  The  effect  of 
the  tension  in  the  conductors  on  the  struc- 
tures in  changing  the  direction  of  the  line  by  this 
method  is  shown  in  Fig.  233.  For  each  conductor 
there  is  a  force  acting  transversely  on  the  structure 

equal  to   2  sin  —  times  the  tension  in  the  conductor.     The 
2 


Fig.  231. 


POWER  TRANSMISSION. 


331 


forces  acting  on  structures  at  curves  should  be  the  same  as 
on  straight  portions.  Therefore  the  spans  on  curves  are 
shortened  by  such  an  amount  that  the  sum  of  the  wind 


Fig.  232. 


pressure  on  the  conductors  and  the  transverse  component 
of  the  tension  shall  be  equal  to  the  wind  pressure  on  the 
conductors  on  straight  portions  of  the  line.  With  shorter 


Fig.  233. 

spans  the  tension  in  the  conductors  is  less,  and  therefore 
the  structures  A  and  F,  Fig.  232,  must  be  guyed  in  order 
to  equalize  the  tensions. 

128.   Example  of  Design  of  Transmission  Line.  —  Let  it  be 

required  to  transmit  5000  kilowatts  a  distance  of  100  miles 
over  a  three-phase  circuit,,  using  aluminum  conductors, 
with  a  voltage  of  66,000  at  the  generating  station,  the 
frequency  being  25. 

ECONOMIC  DROP.  The  formulae  for  the  economic  drop 
and  cross-section  of  conductor  for  a  three-phase  line  are 
given  in  §  1 18  as  equations  (7)  and  (8)  respectively.  Since 
the  conductors  are  to  be  of  aluminum, 

Kl  =  85,000  ohms, 
K2  =  0.0048  pound, 


332  ALTERNATING-CURRENT  MACHINES. 

as  calculated  from  the  constants  given  in  §  123.     Assuming 

ct  =  $15.00, 
.     c2  =  $  0.25, 


then  K  =  4  X  .05  X  .25  X  85,000  X  .0048  X  10,000  X  1000 
=  204,000,000. 

TJ  3  X  204,000,000 

Hence  x  =  -  s2-—  -  —  ^'  -  + 

4  (66,000  )2  X  15 

[9  (204,000,000  )2  +  12  (66,000  )2X  15X204,000,000]* 
4  (66,ooo)2  X  15 

or  the  economic  drop  is  4.61  %  of  the  impressed  voltage. 

If  ^ooo  K.W.  are  to  be  delivered,  then  —  ^  -  =  ^242  K.W. 

i  —  .0461 

must  be  supplied  to  the  line.     The  size  of  the  conductor 
required  is 

c        ifiooo  X  5242  200  1 

5  =  -  --       J    —  85,000  — 

214,356,000,000    ~          .O46ij 

=  221,900  circular  mils. 

An  aluminum  conductor  of  this  size  weighs  221,900  X.0048 
or  1065  pounds  per  mile. 

Proof  that  4.61  %  is  the  economic  drop  for  the   given 
conditions: 

(a)  Annual  cost  of  lost  power  =  242  X  $15  =  $3630. 

(b)  Interest   on    cost   of  line   conductors  =  1065  X  300 

X  $0.25  X  .05  =  $3994. 

(c)  Cost    of    lost    power    per    delivered    kilowatt-year 

-  m%  -  $0.726. 

(d)  Interest   on   cost   of  line   conductors   per  delivered 
kilowatt-year  =  %%%$  =  $0.799. 


POWER   TRANSMISSION.  333 

Hence  the  sum  of  loss  and  interest  per  delivered  kilowatt- 
year  is  $1.525. 

Now  suppose  J-  as  much  conductor  material  were  used. 

Line  loss  =  f  X  242  =  277  K.W. 

Drop  =  f  X  4-61  =  5.27  %. 

Delivered  power  =  5242  —  277  =  4965  K.W. 

Then  the  new  values  of  a,  b,  c,  and  d  are : 

a  =  277  X  $15  =  $4155, 
b  =  I  X  $3994  =  $3498, 
c  =  f  Mf  =  $0.838, 
d  =  |Jff  -  $0.704, 

and  hence  the  sum  of  loss  and  interest  per  delivered 
kilowatt-year  is  $1.542.  Therefore  using  |  as  much  con- 
ductor material  increases  the  cost  of  delivering  power. 
The  cost  of  lost  power  and  interest  on  the  cost  of  the  line 
conductors  have  been  similarly  calculated  for  f ,  J,  and  £ 
as  much  conductor  material.  The  results,  which  are 
plotted  in  Fig.  234,  prove  that  4.61  %  is  the  economic 
drop,  since  the  cost  per  delivered  kilowatt-year  is  a  mini- 
mum at  that  value. 

It  has  been  assumed  that  the  conductor  is  solid  wire; 
but  for  a  conductor  of  this  size,  cable  is  always  used. 
The  resistance  of  a  cable  is  a  few  per  cent  higher  than  that 
of  a  solid  conductor  having  the  same  cross-sectional  area. 
A  cable  of  228,000  circular  mils  has  approximately  the 
same  resistance  as  a  solid  wire  of  221,900  circular  mils. 
A  cable  of  this  size  weighs  noo  pounds  per  mile,  has  a  resist- 
ance of  0.396  ohm  per  mile,  and  is  0.55  inch  in  diameter. 

INDUCTANCE.  —  Assuming  the  conductors  to  be  spaced 


334 


ALTERNATING-CURRENT   MACHINES. 


72  inches  apart,  center  to  center,  the  inductance  per  mile 
of  two  conductors  0.55  inch  in  diameter,  §  120,  is 

2    80.5  +  740  log  -^—   io~e  =  0.00374  henry, 

and  the  inductance  of  the  whole  length  of  the  line,  for  two 
conductors,  is  0.374  henry. 

CAPACITY.      From    §  121,    the    capacity    between    two 

conductors    0.55    inch    in    diameter    is    — — **  =  0.00803 

2.418 

microfarad  per  mile,  and  the  capacity  between  two  con- 
ductors for  the  whole  length  of  line  is  0.803  microfarad. 


DOLLARS  PER  DELIVERED  KW.-YEAR 
-»  £•*  i-*  i-»  M  J- 

i  s  fe  fe  s  a 

/ 

J 

/ 

\ 

/ 

\ 

/ 

V 

/ 

/ 

\ 

/ 

\ 

/ 

^ 

• 

x* 

4  5 

DROP. IN  PER  CENT  OF  IMPRESSED  E.  M,  F. 

Fig-  234- 


REGULATION.  A  convenient  way  of  calculating  the 
regulation  of  a  three-phase  circuit  is  based  upon  the  fact 
that  a  three-phase  circuit  is  equivalent  to  t\vo  single- 


POWER  TRANSMISSION,  335 

phase  circuits  employing  conductors  of  the  same  size.  In 
other  words,  the  regulation  of  a  three-phase  circuit  is  the 
same  as  that  of  a  single-phase  circuit  carrying  half  as 
much  power  with  the  same  percentage  loss,  at  the  same 
voltage  and  distance  between  conductors.  The  induc- 
tance and  capacity  of  a  single-phase  line  with  the  same 
sized  conductors  as  the  three-phase  line  under  considera- 
tion have  just  been  calculated.  Dividing  the  line  into  ten 
equal  sections,  the  constants  of  each  are 

L  =  .0374  henry. 

C  =  .0803  microfarad. 

R  =  .396  X  20  =  7.92  ohms. 

Using  the  notation  of  §  122,  and  assuming  the  voltage  at 
the  receiving  end  of  the  line 

E  =  66,000  (i  —  .0461)  —  62,957  volts. 
If  2500  kilowatts  are  delivered,  the  load  current  is 

T         2,^00,000 

/=^r  =  39-71  amperes. 

The  resistance  and  reactance  drops  of  section  i  are 
respectively 

IR  =  39.71  X  7.92  =  314  volts, 

2  TtfLl    =    50  7T  .0374    X    39.71    =    233   VOltS. 

Hence 

El  =  V (62,957  +  3!4)2  +  (233)'  =  63,271.4  volts. 
The  current  in  section  i  is 

A  =  ^/(39-71)2  +  (507r  63,271.4  X  .ooooooo8o3)2 
=  39.72  amperes. 


336         ALTERNATING-CURRENT   MACHINES. 

Similarly,  the  E.M.F.  across  section  2  is 

£2  =V(63,27i.4  +  39.72  X  7.92)'  +  (50  TT. 0374  X  39-72)2 
=  63,586.0  volts, 

and  the  current  therein  is 


/2  =  ^/(39-72)2+(5°7r63>586X.ooooooo8o3)2  =39. 73  amps. 

Proceeding  in  like  manner,  the  values  of  the  E.M.F. 's  and 
currents  in  each  section  may  be  determined,  and  the  regu- 
lation then  calculated. 

NATURAL  FREQUENCY.  The  inductance  of  the  line  is 
0.374  henry  and  the  capacity  is  0.000000803  farad.  From 
§  115,  the  natural  frequency  is 


4x^0.374  X  .000000803 


=  456  cycles. 


There  is  therefore  no  probability  of  trouble  from  har- 
monics at  the  chosen  frequency  of  25. 

SAG  OF  CONDUCTOR.  The  conductors  are  to  be  strung 
with  such  a  sag  that  at  the  minimum  temperature  with 
one-half  inch  of  ice  all  around  the  cable,  and  a  wind  pres- 
sure of  15  pounds  per  square  foot  of  projected  area,  the 
tension  in  the  cable  shall  not  exceed  the  elastic  limit  of  the 
material  (14,000  pounds  per  square  inch). 

Weight  per  foot  of  cable  is  Hlzj  =  0.208  pound. 

Area  of  conductor  is  0.179  square  inch. 

Outside  diameter  of  cable  is  0.55  inch. 

Since  ice  weighs  57  pounds  per  cubic  foot,  the  weight  of 
an  ice  coating  one-half  inch  thick  is 


I2  TT  - — -^ — -  X  57  =  0.652  pound  per  foot  of  cable. 

1728 


POWER   TRANSMISSION.  337 

The  weight  of  cable  and  ice  is  therefore  0.86  pound  per 
foot  of  cable. 

The  wind  pressure  on  the  ice-covered  cable  is 

15  =  1.94  pounds  per  foot  of  cable. 
144 

The  resultant  of  weight  and  wind  pressure  is 


\/(o.&6)2  +  (i-94)2  =  2.122  pounds  per  foot  of  cable. 

Assuming  a  span  of  400  feet,  then  in  the  notation  of 
§  125, 

Si  =  400. 
W   =  0.86. 

Wr  =   2.122. 

A    =  0.179. 

T   =  14,000  X  0.179  =  2510. 

t   =  o,  75,  and  150. 

k   =  0.0000128. 
E  t  =  9,000,000. 

Hence 

(400  )3  X(2.I22)2 

£'  =  4°°+-     24(2510)'        =401-91' 
40131  -  _.  g. 


8  ] 

8  64  X  9,000,000  X  0.179 

or  D*  —  192  D  =  1585.7 


338  ALTERNATING-CURRENT    MACHINES. 

Solving  by  estimation  and  trial,  the  sag,  D,  is  found  to  be 
16.9  feet. 


The  vertical  sag  D'  =  --  X  16.9  =  6.84  feet. 

2.122 

If  t  =  75,  there  results 

D3  -  3  Xo4°°  [401.28  (i  +  0.00096)  -  400]  D  =  1585.7, 
8 

from  which  D  =  18.35  ^eet- 

The  vertical  sag  =  —  —  X  18.35  =  7.43  feet. 

2.122 

If  /  =  150,  then 

D3-3  X  04°0  [401.  28  (i  +  0.00192)-  400]  Z>=  1585.7; 
8 

.'.  D  =  19.69  feet. 

Vertical  sag  =  —  -  —  X  19.69  =  7.98  feet. 
2.122 

If  the  minimum  temperature  is  taken  as  —40°  F.,  75° 
above  the  minimum  is  35°  F.,  and  150°  above  the  mini- 
mum is  110°  F.  In  Fig.  235  the  vertical  sags  for  spans  of 
400  feet,  500  feet,  600  feet,  and  700  feet  have  been  plotted 
in  terms  of  temperatures  between  —40°  and  no°F.  In 
stringing  the  cables,  the  proper  sag  to  be  allowed  should 
be  obtained  from  these  curves,  its  value  depending  upon 
the  temperature  at  that  time. 

LENGTH  OF  STANDARD  SPAN.  From  the  curves  of 
Fig.  235,  the  lower  curve  of  Fig.  236  has  been  drawn, 
showing  the  vertical  sag  at  110°  F.  for  different  span 
lengths.  If  the  minimum  clearance  of  the  cables  from 
the  ground  is  to  be  20  feet,  the  point  of  support  of  the 


POWER   TRANSMISSION 


339 


cables  must  be  at  a  distance  from  the  ground  equal  to  20 
feet  plus  the  maximum  sag.  The  distance  of  the  point  of 
support  of  the  lowest  cable  from  the  ground  is  called,  for 


22 
20 

18 
16 

ti 

s 

I14 

CO 

10 
8 

6 

-41 

,i 

~ 

— 

=« 

— 

70C 

.-     - 

IFT^ 

•      "- 

_      - 

— 

— 

60C 

FT.  ' 

PAN 

- 

- 

—      — 

• 

—     - 

— 

.-  -  -•• 

i—  —  ' 

500 

EI£ 

'AN- 

— 

— 

•       - 

- 

; 

4 

DO  FT 

.SPM 

- 

\ 

3°           -20U            0°              20°           40°             60°             80°            100° 

TEMPERATURE  (FAHR.) 

Fig-  235- 

convenience,  the  height  of  the  tower.  The  upper  curve  of 
Fig.  236  has  been  drawn  with  ordinates  representing  20 
feet  more  than  those  of  the  lower  one,  and  therefore  shows 


340 


ALTERNATING-CURRENT   MACHINES. 


the  heights  of  towers  for  different  span  lengths.     Assume 
that  66,000-volt  insulators  cost  $5.00  each  erected,  and  that 


500  600 

SPAN  LENGTH  IN  FEET 
Fig.  236. 


the   costs  of   towers  of   various   heights,  erected,   are  as 
follows : 


Tower  Height  in  Feet. 

Cost  of  Tower  in 

Dollars. 

30 

95 

32-5 

100 

35 

no 

37-5 

125 

40 

*45 

From  the  upper  curve  of  Fig.  236,  it  is  seen  that  the 
greatest  span  length  for  which  a  30-foot  tower  can  be  used 
under  the  given  conditions  is  450  feet.  With  this  span 


POWER  TRANSMISSION.  341 

length  there  will  be  required  11.73  towers  per  mile,  and 
the  cost  of  towers  and  insulators  per  mile  of  line  will  be 

11.73  X  $95  =  $iii4-35  for  towers. 
3  X  11.73  X  $5  =      175.95  for  insulators. 

$1290.30  =  total  cost  per  mile. 

For  32.5-foot  towers  the  span  is   515  feet,  and   10.25 
towers  are  required  per  mile.     Then 

10.25  X$ioo  =  $1025.00  for  towers. 
3  X  10.25  X  $5  =      153.75  for  insulators. 

$1178.75  =  total  cost  per  mile. 

For  35-foot  towers  the  span  is  570  feet,  and  9.26  towers 
are  required  per  mile.     Then 

9.26  X  $no  =  $1018.60  for  towers. 
3  X  9.26  X  $5  =      138.90  for  insulators. 

$1157.50  =  total  cost  per  mile. 

For  3 7. 5 -foot  towers  the  span  is  615  feet,  and  8.6  towers 
are  required  per  mile.     Then. 

8.6  X  $125  =  $1075.00  for  towers. 
3  X  8.6  X  $5  =      129.00  for  insulators. 

$1204.00  =4total  cost  per  mile. 

For  40-foot  towers  the  span  is  665  feet,  and  7.94  towers 
are  required  per  mile.     Then 

7.94  X  $145  =  $1151-30  for  towers. 
3  X  7.94  X  $5  =      119.10  for  insulators. 

$1270.40  =  total  cost  per  mile. 

It  is  evident  from  the  foregoing  calculations  that  the 
economic  span  is  570  feet,  employing  35-foot  towers. 
FORCES  ACTING  ON  THE  TOWERS.    For  spans  of  570  feet, 


342  ALTERNATING-CURRENT  MACHINES. 

the  force  acting  on  each  tower  due  to  the  weight  of  line 
conductors  when  covered  with  ice  will  be 

3  X  570  X  0.86  =  1470.6  pounds. 

The  pressure  due  to  the  wind,  being   15   pounds   per 
square  foot  of  projected  cable  area,  when   the  cable  is 
covered  with  one-half  inch  of  ice  all  around,  is 
3  X  570  X  1.94  =  33J7-4  pounds. 

The  weights  of  towers  vary  considerably,  depending 
upon  their  design.  One  ton  may  be  taken  as  the  average 
weight  of  a  35-foot  tower. 

A  tower  of  the  size  under  consideration  will  have  the 
equivalent  of  about  25  square  feet  of  normal  surf-ace 
exposed  to  the  wind.  Hence  the  wind  pressure  on  the 
tower  is  25  X  30  =  750  pounds.  This  acts  at  the  center 
of  gravity  of  the  exposed  surface,  but  for  the  purpose  of 
calculation  it  is  assumed  that  half  this  force,  375  pounds, 
acts  at  the  top  of  the  tower. 

Therefore  the  tower  must  be  strong  enough  to  resist  a 
force  of  1470  +  2000  =  3470  pounds  acting  vertically 
downward,  and  a  force  of  3317  +  375  =  3692  pounds 
acting  horizontally  at  the  top  of  the  tower. 

LENGTH  OF  SPAN  ON  CURVES.  Where  the  line  is  carried 
around  a  curve  as  shown  in  Fig.  232,  the  transverse  force 
acting  on  the  tower  due  to  the  tension  in  the  cables 
should  be  allowed  for  by  shortening  the  span  length. 
If  the  angle  a  be  2°,  the  transverse  force  due  to  the  tension 
in  the  cables  (Fig.  233)  is 

3  X  2  X  2510  X  sin  i°  =  263.5  pounds. 

The  transverse  force  due  to  wind  pressure  on  the  con- 
ductors is  3317.4  pounds  in  the  standard  span.  Sub- 
tracting 263.5  therefrom  leaves  3054  pounds  as  the 


POWER  TRANSMISSION.  343 

desired  wind  pressure  on  the  conductors  per  span  on  the 
curve.     Hence  the  length  of  such  spans  should  be 

3°^4  X  570  =  525  feet. 
oo   / 

If  the  angle  «  be  4°,  then  the  transverse  force  due  to 
the  tension  in  the  cables  is  525.6  pounds.  The  span  length 
for  this  value  of  a.  is 

33I7.4-525.J  8Qfeet 


Similarly,  when 

a  =    6°,  the  span  is  432  feet, 

a  =    8°,  the  span  is  390  feet, 

«  =  10°,  the  span  is  345  feet. 

It  is  not  advisable  to  have  the  angle  a  greater  than  10°. 
If  too  many  towers  will  then  be  required  to  make  the 
necessary  turn,  it  is  better  to  make  it  as  shown  in  Fig.  231, 
by  dead-ending  the  line  on  two  towers  and  having  a  short 
slack  span  between  them,  rather  than  by  means  of  a  curve. 

PROBLEM. 

Thirty  thousand  kilowatts  are  to  be  transmitted  over  a  section  of  a 
transmission  line  53  miles  long,  using  a  three-phase  circuit  of  alumi- 
num conductors,  with  .110,000  volts  at  the  generating  station.  The 
various  constants  are: 

Frequency  =  25. 

Cost  of  power  per  kilowatt-year  at  generating  station  =  $12.00. 

Cost  of  aluminum  per  pound  =  $0.25. 

Interest  rate  thereon  =  4  %. 

Distance  between  cables  =  9  feet. 

(Fig.  230  shows  the  type  of  towers  used.) 

Determine  the  economic  drop,  cross-section  of  conductor,  natural 
frequency  of  the  line,  and  the  charging  current  per  conductor.  Pre- 
pare curves  showing  the  vertical  sag  at  different  temperatures  for 
various  span  lengths. 


INDEX. 


[The  figures  refer  to  page  numbers.] 


Addition  of  vectors,  17. 
Admittance  of  circuit,  72, 

representation  of,  74, 
Admittances,  polygon  of,  87. 
Ageing  of  iron,  188. 
Air-blast  transformers,  194. 
Air  gap  of  induction  motors,  227. 
Alexanderson  alternator,,  145. 
All-day  efficiency,  167,, 
Alternating  current,  definition  of,  i. 

power  transmission,  300. 
Alternations,  definition  of,  i. 
Alternator,  94. 

compensated,  128, 

efHciency  of,  133. 

flux  in,  117. 

General  Electric  Co.'s,  125,  128, 
141. 

inductor  type,  136. 

losses  in,  134. 

rating  of,  135. 

regulation,  116. 

revolving-field  type,  139. 

saturation  curves,  114. 

self-exciting,  145. 

Stanley,  137. 

voltage  drop  in,  117. 

voltage  of,  96. 

Westinghouse,  126. 
Alternators  in  parallel,  262. 
Aluminum  line  wire,  319. 
Angle  of  hysteretic  advance,  160. 
Angle  of  lag  or  lead,  12,  71. 


Apparent  resistance,  38,  72. 
Armature  copper  loss,  120,,  134. 

E.M.Fc  generated  in,  96." 
Armature  impedance  voltage,  121. 

inductance,  119. 

reaction  of  converters,  291., 

resistance  drop,  121. 

windings,  99. 
Autotransformer,  151. 

connections  of,  186. 
Average  value  of  current  and  pres- 
sure, 8. 

Balanced  polyphase  systems,  105. 
Belt  leakage  reactance,  224. 

Calculation  of  alternator  regulation, 

119. 

induction  motor  leakage  react- 
ance, 216 
resultant  admittance,  89,  91. 

impedance,  86,  91. 
transformer  leakage  inductance, 

168. 

Capacity,  distributed,  303. 
formulae,  53. 
of  condensers,  50. 
of  transmission  lines,  313,  334. 
reactance,  64,  72. 
unit  of,  51. 

Centrifugal  clutch  pulley,  210. 
Charging    current    of    transmission 
line,  315. 


345 


346 


INDEX. 


Choke  coils,  44. 

Circle  diagram  of  induction  motor, 

231,  233. 

of  transformer,  179. 
Circuits,  natural  period  of,  80,  83. 

time  constant  of,  34. 

with  R,  L,  and  C,  70. 
Coefficient,  leakage,  233. 

of  self-induction,  27. 
Coil-end  leakage  reactance,  222. 
Compensated  alternators,  128. 

series  motors,  270. 
Compensators,  connections  of,  186. 

synchronous,  257. 
Complex  numbers,  representation  of 

Z  and  Y  by,  74. 
Composite  winding,  125. 
Concentrated  armature  "windings,  99. 
Condenser,  capacity  of,  50. 

compensator,  241. 

construction  of,  51. 

hydraulic  analogy,  60. 

resistance,  52. 
Condensers,  48. 

in  parallel  and  in  series,  55. 
Condensive  circuit,   phase   relations 

in,  61. 

Conductance  of  circuit,  73. 
Conductive  compensation,  271. 
Connections  of  transformers,  181. 
Constant-current  transformers,  195. 

potential,  regulation  for,  124. 
Converter,  284. 

armature  heating,  290. 
reaction,  291. 

capacity,  291. 

coils,  current  in,  290. 

current  relations  in,  288. 

E.M.F.  relations  in,  286. 

hunting  of,  293. 

inverted,  285. 

mercury  vapor,  296. 


Converter,  regulation  of,  293. 
split-pole,  296. 
starting  of,  291. 
Cooling  of  transformers,  192. 
Copper  line  wire,  319. 
loss  in  transformers,  165. 
of  armature,  120,  134. 
Core  flux  of  transformers,  154. 
loss  in  transformers,  156. 
-type  of  transformer,  149,  191. 
Counter  E.M.F.  of  self-induction,  27. 
Critical  frequency,  81. 
Current    and    voltage    relations    in 

condensive  circuit,  63= 
in  polyphase  systems,  101,  104, 

1 06. 

average  value  of,  8. 
components  of,  16,  159,  227. 
effective  value  of,  7. 
flow,  expression  for,  71. 
instantaneous,  in  alternating-cur- 
rent circuits,  41,  65,  76. 
values  of,  3. 
lag  or  lead  of,  12. 
magnetic  energy  of  started,  36. 
produced    by    harmonic    E.M.F. , 

37,  64. 

relations  in  converters,  288. 
Currents,     single-phase    and    poly- 
phase, 13. 
Curve,  efficiency,  of  alternator,  135. 

transformer,  167. 
non-sine,  form  factor  of,  10. 
saturation,  114. 
sine,  4. 

form  factor  of,  9. 
Curves  in  transmission  line,  330. 
Cycle,  definition  of,  i. 

Damped  oscillations,  82. 

effective  current  value  oi,  83 
Damping  factor,  82. 


INDEX. 


347 


Decaying  currents,  34,  57. 

oscillatory  current,  82. 
Decrement  of  oscillations,  82. 
Definition  of  terms,  71. 
Delta  connection,  100,  184. 
Design  of  transmission  line,  331. 
Dielectric  constants,  52. 

energy  stored  in,  60. 

for  condensers,  51. 

hysteresis,  52. 

polarization,  E.M.F.  of,  58. 

strength  of  materials,  50. 
Direct-current   power   transmission, 

300. 
Distance    between  line  conductors, 

313- 
Distortion  of  E.M.F.  wave,  causes 

of,  5- 
Distributed  capacity,  303. 

windings,  97,  101,  205,  272. 
Distribution  constant,  98. 
Drop    of    voltage    in    transmission 
lines,  306,  331. 

Economic  drop  in  line,  306,  331. 
Eddy  current  loss  in  induction  mo- 
tors, 228. 
transformers,  157. 
Effective    values    of     current    and 

pressure,  7.         . 
Efficiency,  all-day,  167. 
curve  of  alternators,  135. 
of  alternators,  133. 
induction  motors,  214. 
transformers,  166. 
E.M.F.,  average  value  of,  8. 
counter,  of  self-induction,  27. 
effective  value  of,  7. 
generated  in  armature,  96. 
instantaneous  value  of,  5,  24, 
methods  of  calculating  alternator 
regulation,  119. 


E.M.F.,  of  dielectric  polarization,  58. 

synchronous  motor,  256. 
relations  in  converters,  286. 
wave,  shape  of,  6. 
E.M.F.'s  in  series,  20. 

of  plain  series  motor,  265. 
Electrose  insulators,  321. 
Electrostatic  capacity,  see  Capacity. 
Energy  of  a  started  current,  36. 

stored  in  dielectric,  60. 
Equivalent  R,  X,  and  Z  of  trans- 
former, 163. 
rotor  resistance,  235. 
sine  wave,  definition  of,  18. 
Exciting  current  of  induction  motor, 

227,  233. 

transformer,  151,  159. 
Expression  for  current  flow  in  any 
circuit,  71. 

Farad,  definition  of,  51. 

Field,  rotating,  202. 

Flux  density  in  induction    motors, 

229. 

transformers,  155. 
fringing  constant,  222. 
Forced  compensation,  271. 
Form  factor,  definition  of,  9. 
of  non-sine  curves,  10. 

sine  curve,  9. 
Formulae  for  calculating  capacities, 

S3- 

for  calculating  inductances,  31. 
Four-phase  systems,  106. 
Fractional-pitch     motor     windings 

218. 

Frequencies,  standard,  2,  302. 
Frequency  and  speed,  2. 
changers,  244. 

for  power  transmission,  302. 
natural,  of  transmission  lines,  303. 
resonant,  80. 


348 


INDEX. 


Full-load  saturation  curve,  114. 
-pitch  motor  windings,  218. 

Gauss,  definition  of,  28. 
General    Electric    Co.'s    alternator, 
125,  128,  141,  145. 

induction  motor,  204. 

motor  starter,  208. 

regulator,  295. 

synchronous  motor,  259. 

transformer,  190,  194. 
Growth  of  current  in  inductive  cir- 
cuit, 33. 

Harmonic  shadowgraph,  3. 
Harmonics  of  fundamental  E.M.F., 

23- 

Heating  of  converter  coils,  290. 
Henry,  definition  of,  27, 
Hunting  of  converters,  293. 

synchronous  motors,  256. 
Hydraulic  analogy  of  condenser,  60. 
Hysteresis,  dielectric,  52. 

loop,  162. 

loss  in  induction  motors,  228. 
Hysteresis  loss  in  transformers,  158. 
Hysteretic  advance,  angle  of,  160. 

constant,  158. 

Ideal  transformer,  151. 

vector  diagram  of,  154. 
Impedance,  definition  of,  38. 

of  circuit,  72. 

representation  of,  74. 

synchronous,  117. 

voltage,  armature,  121. 
Impedances  in  series  and  in  parallel, 
90. 

polygon  of,  83. 
Inductance,  armature,  119. 

formulae  for,  31. 

of  transmission  lines,  310,  333. 

practical  values  of,  29. 


Inductance,  self,  described,  260 

unit  of  self,  27. 
Induction  motor,  202. 
air  gap  of,  227. 
calculation  of  exciting  current, 

227. 

of  leakage  reactance,  216. 
circle  diagram  of,  231,  233. 
efficiency  of,  214,  241. 
exciting  current  of,  233. 
flux  density  in,  229. 
General  Electric  Co.'s,  204. 
leakage  coefficient  of,  233. 
losses  in,  240. 

magnetizing  current  of,  230. 
performance  curves,  236. 
power  factor  of,  232. 
resistance  of  windings,  235. 
rotors  of,  205. 
single- phase,  242. 
slip  of,  210,  241. 
speed  and  efficiency,  213. 

regulation,  245. 
starting  of,  207,  244. 
test  with  load,  238. 
torque  and  slip,  212,  226, 
torque  of,  214. 

transformer    method   of    treat- 
ment, 215. 
Westinghouse,  204. 
windings,  205,  218. 
wattmeter,  246. 
Inductive  compensation,  272. 

reactance,  38,  72. 
Inductor  alternators,  136. 
Instantaneous    current   in    alternat- 
ing-current circuits,  41,  65,  76. 
values  of  current  and  voltage  4, 

24. 

Insulators,  319. 
Interpretation  of  symbol  /,  75. 
Inverted  converter,  285. 


INDEX. 


349 


Lag  or  lead  of  current,  12,  71. 
Leakage  coefficient,  233. 

reactance    of    induction    motors, 

216. 

transformers,  168. 
Lighting  transformers,  188. 
Lightning  arrester  choke  coils,  46. 
Line  capacity,  313,  334. 

constants,  319. 
Line  inductance,  310,  333. 
natural  frequency  of,  303. 
resistance,  309. 
structure,  326. 
wire,  cross-section  of,  307. 
material,  318. 
relative  weights  of,  305. 
sag  of,  322,  336. 
wires,  distance  between,  313. 

wind  pressure  on,  323. 
Linkages  defined,  27. 
Load  losses  in  alternators,  134. 
saturation  curve,  114. 
test  on  induction  motors,  238. 
Logarithmic  change  of  current,  34. 

decrement  of  oscillations,  82. 
Losses  in  induction  motors,  240. 
synchronous  machines,  134. 
transformers,  156. 

Maclaurin's  series,  76. 

Magnetic  energy  of  started  current, 

36. 

flux  in  alternators,  117, 
leakage  in  induction  motor,  216. 

transformer,  168. 

Magnetizing    current    of    induction 
motor,  230. 
transformer,  159. 
wave  of  transformer,  162. 
M.M.F.  method  of  calculating  alter- 
nator regulation,  123. 
Magnitude  of  self-induction,  30. 


Material  of  line  conductors,  318. 
Maxwell,  definition  of,  28. 
Measurement  of  power,  107. 
Mercury  vapor  converter,  296. 
Mesh     or     delta    connection,    100, 

184. 

Microfarad,  definition  of,  51. 
Monocyclic  system,  244. 
Motor,  induction,  see  Induction  mo- 
tor. 

repulsion,  277. 
series-repulsion,  280. 

single-phase,  see  Series  motor, 
starters,    General    Electric    Co.'s, 

207. 

Westinghouse,  209. 
synchronous,      see      Synchronous 
motor. 

Natural  draft  transformers,  192. 
frequency   of   transmission   lines, 

3°3- 

period  of  circuit,  80,  83. 
No-load  saturation  curve,  114. 
Non-sine  curves,  form  factor  of,  10. 
phase  difference  of,  18. 

Obstructance,  definition  of,  49. 
Oil-cooled  transformers,  194. 
Operation  of  induction  motors,  210. 
Operative     range    of    synchronous 

motors,  253. 
Oscillations,  damped,  82. 

Parallelogram  of  E.M.F.'s,  21. 
Parallel    operation    of     alternators, 

262. 

Percentage  of  saturation,  114. 
Performance    curves     of     induction 

motor,  236. 
Phase,  12. 

-belt  of  conductors,  223. 


350 


INDEX. 


Phase,  difference  of  non-sine  curves, 

18. 

sine  curves,  12. 
or  distribution  constant,  98. 
relations    in    condensive   circuit, 

61. 
Phases,  number  of,  for  transmission, 

3°4- 
Phase  splitters,  241. 

-wound  rotors,  206. 
Pin-type  insulators,  320. 
Pitch   factor  of    induction    motors, 

218. 
Plain  series  motor,  264. 

characteristics  of,  268. 
Polygon  of  admittances,  87. 
E.M.F.'s,  22. 
impedances,  83. 
Polyphase  alternators,  94. 
currents,  14. 

power,  measurement  of,  109. 
transformers,  198. 
Porcelain  insulators,  320. 
Power   component   of   current,    16, 

159,  227. 

factor,  definition  of,  17. 
of  induction  motor,  232. 
of  three-phase  balanced  circuits, 

112. 

of  transmission  lines,  318. 
in  alternating-current  circuits,  14. 
measurement  of,  107. 
transmission,  frequency  for,  302. 
number  of  phases  for,  304. 
systems  of,  300. 
voltage  for,  305. 
Pressure,  average  value  of,  8. 
Pressure  curves,  actual,  6. 

distortion  of,  5. 
effective  value  of,  7. 
for  power  transmission,  305. 
instantaneous  value  of,  4. 


Preventive  leads,  276. 

Primary  of  induction  motor,  216. 

transformer,  149. 

Problems,  25,  47,  68,  92,  146,  200, 
282,  299,  343, 

Quarter-phase  currents,  13. 
systems,  101. 

Radius  vector,  5. 
Rating  of  alternators,  135. 
Ratio  of  transformation,  149. 
Reactance  of  any  circuit,  72. 
condensive  circuit,  64. 
inductive  circuit,  38. 
Reactors,  44. 

Rectifier,  mercury  vapor,  296. 
Regulation    for    constant    potential, 

124. 
of  alternators,  116. 

methods  of  calculating,  119. 
of  converters,  293. 
of  induction  motors,  245. 
of  transformers,  173,  181. 
of  transmission  line,  316,  334. 
Regulator,  General  Electric  Co.'s, 

295- 

Stillwell,  294. 
Tirrill,  130. 
Reluctance    of     transformer     core, 

160. 

Representation  of  Z  and  Y  by  com- 
plex numbers,  74. 
Repulsion  motor,  277. 

starting  of  single-phase  induc- 
tion motor,  279. 
series  motor,  280. 
Resistance,  apparent,  38,  72. 
drop,  armature,  121. 
leads  for  series  motors,  276. 
of  line  wire,  309. 
Resonance,  80. 


INDEX.  351 

Resultant  admittance,  89.  Single-phase  alternators,  94. 

E.M.F.  of  harmonic  components,  commutator  motors,  262. 

23.  current,  13. 

impedance,  86.  induction  motor,  242,  ^^gr1 

Revolving-field  type  alternators,  139.  Skin  effect  of  wire,  45,  310.^ 

Rotary  converter,  see  Converter.  Slip  of  induction  motors,  210,  241. 

Rotating  magnetic  field,  202.  Slot  contraction  factor,  225. 

Rotor  of  induction  motor,  203.  leakage  reactance,  217. 

phase- wound,  206.  Solenoids,  self-inductance  of,  32. 

squirrel-cage,  205.  Span  lengths  on  curves,  342. 

Spans   on   transmission   lines,    329, 

Sag  of  transmission  lines,  322,  336.  338. 

Saturation,  113.  Sparking  in  series  motors,  274. 

curves  of  alternator,  114.  Split-pole  converter,  296. 

factor,  115.  Squirrel-cage    motors,    starting    of, 

Scott  transformer,  183.  207. 

Secondary  of  induction  motor,  216.  rotors,  205. 

transformer,  149.  Standard  frequencies,  2. 

Self-exciting  alternator,  145.  Stanley  alternator,  137. 

-inductance,  counter    E.M.F.   of,  transformer,  192. 

27,  37.  Star  or  F-connection,  100,  184. 

described,  26.  Started  current,  magnetic  energy  of, 

formulae  for,  31.  36. 

unit  of,  27.  Starting  converters,  291. 

Series  motor,  compensated,  270.  induction  motors,  207,  244,  279. 

characteristics  of,  273.  synchronous  motors,  258. 

connections  of,  272.  Stator  of  induction  motor,  203. 

performance  curves,  275.  windings,  205. 

plain,  264.  Step-up    and    step-down     transfer- 
characteristics  of,  268.  mation,  150. 
E.M.F.'s  of,  265.  Stillwell  regulator,  294. 
resistance  leads  of,  276.  Strength  of  dielectrics,  50. 
sparking  in,  274.  Structures,  transmission  line,  326. 
Westinghouse compensated,  271.  Susceptance  of  circuit,  73. 
-repulsion  motor,  280.  Suspension-type  insulators,  322. 
Shading  coil,  248.  Synchronizer,  258. 
Shadowgraph,  harmonic,  3.  Synchronous  compensators,  257. 
Shell-type  of  transformer,  149,  192.  converter,  see  Converter. 
Sine  curve,  4.  impedance,  117. 

form  factor  of,  9.  machines,  losses  in,  134. 

wave,   equivalent,     definition    of,  motor,  249. 

18.  behavior  of,  252. 


352  INDEX. 

Synchronous  motor:  Transformer,  flux  in,  154. 

efficiency  of,  255.  for  lighting,  188. 

E.M.F.,  256.  General  Electric  Co.'s,  190,  194, 

General  Electric  Co.'s,  259.  197. 

hunting  of,  256.                         .  graphic  representation  of,  151. 

operative  range  of,  253.  hysteresis  loss  in,  158. 

stability  of,  255.  ideal,  151. 

starting,  258.  vector  diagram  of,  154. 

V-curves  of,  258.  losses,  156. 

Synchroscope,  260.  magnetizing  current  of,  159. 

wave,  162. 

Table  of  converter  capacities,  291.  method  of  induction  motor  treat- 
dielectric  constants,  52.  ment,  215. 
strengths,  50.  oil-cooled,  194. 
line  constants,  319.  polyphase,  198. 
Temperature  effect  on  core  loss,  159.  regulation  of,  173,  181. 
Three-phase    power    measurement,  Scott,  183. 

no.  Stanley,  192. 

systems,  104.  vector  diagram  of,  175,  178. 

transformations,  184.  Wagner,  188. 

Thury   system   of   power   transmis-  water-cooled,  194. 

sion,  300.  Westinghouse,  191. 

Time  constant  of  circuit,  34,  59.  with  divided  coils,  172. 

Tirrill  regulator,  130.  Transmission  line,  see  also  Line. 

Tooth-tip  leakage  reactance,  220.  charging  current  of,  315. 

Torque  of  induction  motors,  214.  design  of,  331. 

Transformation,  ratio  of,  149.  economic  drop  in,  306,  331. 

Transformer,  air-blast,  194.  natural  frequency,  303,  336. 

calculation   of   leakage   reactance  power  factor  of,  318. 

of,  168.  regulation,  316,  334. 

circle  diagram  of,  179.  span  lengths,  329,  338. 

connections,  181.  structures,  326. 

^constant-current,  195.  forces  acting  on,  341. 

cooling  of,  192.  of  power,  300. 

copper  losses,  165.  Triangle  of  E.M.F.'s,  38,  64. 

core  losses,  156.  Two-phase  power  measurement,  107. 

reluctance  of,  160.  systems,  101. 
definitions,  149. 

eddy  current  loss  in,  157.  Values  of  inductances,  29. 

efficiency,  166.  Vapor  converter,  mercury,  296. 

equivalent  R  and  X  of,  163.  V-curves     of    synchronous     motor, 

exciting  current  of,  151,  159.  2580 


INDEX. 


353 


Vector,  5. 

addition  and  subtraction,  17. 
diagram  of  transformer,  154,  175, 

178. 
Voltage  and  current  relations  in  con- 

densive  circuit,  63. 
in  polyphase  systems,  101,  104, 

1 06. 

armature  impedance,  121. 
average  value  of,  8. 
curves  of  actual,  6. 
drop  in  alternators,  117. 

transmission  line,  306,  331. 
effective  value  of,  7. 
for  power  transmission,  305. 
generated  in  armature,  96. 

Wagner  single-phase  motor,  279. 

transformer,  188. 
Water-cooled  transformer,  194, 


Wattless  component  of  current,  16, 

159,  230. 

Wattmeter,  induction,  246. 
Wave-shape,  3. 

causes  of  distortion  of,  5. 

determination  of  form  factor  of, 
10. 

of  current  in  converter  coils,  290. 
Weights,  relative,  of  line  wire,  305. 
Westinghouse  alternator,  126. 

compensated  series  motor,  271. 

induction  motor,  204. 

motor  starter,  209. 

transformer,  191. 
Wind  pressure  on  lines,  323. 

Y-connection,  100,  184. 
of  compensators,  186. 

Zig-zag  leakage  reactance,  220. 


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